Black Holes & Qubits

Transcription

1 Black Holes & Qubits Duminda Dahanayake Imperial College London Supervised by Mike Duff Working with: Leron Borsten & Hajar Ebrahim

2 Essential Message Stringy Black Hole Entropy = Quantum Information Theory Entanglement Measure

3 Outline I. Introduction Stringy BH Entropy Quantum Information Entropy-Entanglement Correspondence II. Work in progress Freudenthal Triple System Cayley over Imaginary Octonions Summary

4 Part I Introduction

5 Black Hole Side BH entropy is expressed in terms of electric (q) and magnetic (p) charges. Charges arise from wrapping of D-branes on cycles in a Calabi-Yau compactification space. Axion Dilaton BH 4 charges Simple determinant-like entropy expression Will be related to a 2 qubit system

6 N = 2, D = 4 STU model (SUGRA) BH Reduction of type IIA on CY with D0, D2, D4, D6 8 charges More complicated entropy expression: Will be related to a 3 qubit system.

7 N = 8 SUGRA / M-theory BH Most general BH / Black ring entropy is related to the Cartan E 7(7) quartic invariant I charges this time X and Y are antisymmetric matrices made up of the charges. Pf is the Pfaffian: the square root of the determinant of an antisymm matrix. Very large expression when written out in full. After an SU(8) transformation of the charge matrices to bring them into a canonical form, the system is shown to be similar to the 3 qubit system, but corresponds to 7 tripartite entangled qubits.

8 QI Side A qubit is a 2-level quantum system: Qutrits have 3 levels, qu-4-its have 4 levels,..., and qudits have n-levels. An N qudit system is a linear combination of N-fold tensor products: For a pure 2-qubit system the entanglement E is given by the von Neumann entropy (extension of Shannon entropy):

9 A related entanglement measure is the concurrence, C. C is a nonlinear (actually transcendental) function of E, but E increases monotonically from 0 to 1 as C goes from 0 to 1. For mixed states the concurrence is more complicated. The entanglement measure for a 3-qubit system is the 3-tangle (C is the square-root of the 2-tangle). This is proportional to Cayley s hyperdeterminant: Det: HyperDet:

10 Can write as a determinant of determinant-like quantities: A more schematic representation: Another, more detailed, schematic representation: If the s were distinct, there would be 64 = 2 6 terms; as it is, there are fewer. Q: how many ways are there to hook up indices so that no has it s own indices linked?

11 Correspondence One can set up one-to-one correspondences between BH charges and state-vector coefficients. (We ll refer to the correspondences as dictionaries ). Axion Dilaton: STU model: N = 8: Think big...

12 Part II Work in Progress

13 Octonions They are: An 8-dim vector space, a nicely normed division algebra, a -algebra, a Jordan algebra, noncommutative, non-associative, alternative, etc. Imaginary octonions have ~antisymmetric multiplication Octonion multiplication table: Canonical table generated by Cayley-Dickson process. Contains quaternion and binarion tables. The Cayley-Dickson construction generates new - algebras from old ones using this definition of the product:

14 Split octonions are generated by a different recursive rule: Structure constants: ordinary split

15 Freudenthal The Freudenthal triple system (FTS) is used to construct the 56 dimensional rep of E 7 and we can use it to calculate the quartic invariant I 4. The J 3 s are octonion valued 3 3 Hermitian matrices: Capital P s and Q s are general octonions, while lowercase p s and q s are real.

16 Defining I 3 = Det[J 3 ] (where choices are made about ordering) we obtain the quartic invariant as: Symmetric Trace Jordan product Quadratic adjoint map When expanded, the expression is still very large, but can be calculated quickly. Excluding charges restricts to different subsectors.

17 Keeping only the real octonions (lowercase charges) reproduces the STU model entropy. Keeping only one of the 3 pairs P x, Q x we obtain N = 4 subsectors. I 4 is related to Cayleys hyperdeterminant via an SU(8) transformation, but at the moment we re trying to relate pure Q,P subexpressions of I 4 to Cayleylike structures using simple dictionaries. cf

18 The dictionary we thought was the best bet doesn t match, so we tried all permutations of it. All 40,320 of them. There were still no matches though (and only 1536 give the right number of terms). We then tried other Cayley structures:,,,... There are as many Cayley structures as there are ways of picking 6 sets of 2 from 4 sets of 3, such that no set of 2 contains elements from the same set of 3 (order doesn t matter). That s 3348, by the way = 5,142,528. There were still no matches, so now we think we need a new dictionary.

19 Cayley over Im O Another aspect we re looking at is writing I 4 as Cayley s hyperdeterminant over the imaginary octonions. Making the state vector coefficients octonion valued is complicated by the non-commutativity and nonassociativity of octonion multiplication. If we choose X ijkl = ij kl we get agreement with the quaternionic subsectors (but only using different dictionaries for each), but we re still struggling with the unrestricted case; perhaps C ijkl associator coefficients? We suspect more Cayley structures may be needed to obtain agreement, perhaps by adding quantities that are only nonzero in the octonionic case.

20 Summary There exists a tantalising BH entropy QI entanglement correspondence The Freudenthal construction shows great promise as a way to explore this link For now, we need to refine our dictionaries to match I 4 to Cayley structures Can we use the correspondence to clarify entanglement for other qudit systems? E 8 has an SL(2) 8 subgroup and a quintic invariant...

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