Hyperdeterminants as integrable 3D difference equations (joint project with Sergey Tsarev, Krasnoyarsk State Pedagogical University) Thomas Wolf Brock University, Ontario Ste-Adèle, June 26, 2008 Thomas Wolf Hyperdeterminants: a CA challenge I Modern applications of hyperdeterminants I The classical heritage: A.Cayley et al. I The definition of hyperdeterminants and its variations I The next step: the (a?) 2 × 2 × 2 × 2 hyperdeterminant (B.Sturmfels et al., 2006) I FORM computations: how far can we reach now? I Principal Minor Assignment Problem (O.Holtz, B.Sturmfels, 2006) I Hyperdeterminants as discrete integrable systems: 2 × 2 and 2 × 2 × 2. I Is 2 × 2 × 2 × 2 hyperdeterminant integrable?? Plan: I The simplest (hyper)determinants: 2 × 2 and 2 × 2 × 2. Thomas Wolf Hyperdeterminants: a CA challenge I The classical heritage: A.Cayley et al. I The definition of hyperdeterminants and its variations I The next step: the (a?) 2 × 2 × 2 × 2 hyperdeterminant (B.Sturmfels et al., 2006) I FORM computations: how far can we reach now? I Principal Minor Assignment Problem (O.Holtz, B.Sturmfels, 2006) I Hyperdeterminants as discrete integrable systems: 2 × 2 and 2 × 2 × 2. I Is 2 × 2 × 2 × 2 hyperdeterminant integrable?? Plan: I The simplest (hyper)determinants: 2 × 2 and 2 × 2 × 2. I Modern applications of hyperdeterminants Thomas Wolf Hyperdeterminants: a CA challenge I The definition of hyperdeterminants and its variations I The next step: the (a?) 2 × 2 × 2 × 2 hyperdeterminant (B.Sturmfels et al., 2006) I FORM computations: how far can we reach now? I Principal Minor Assignment Problem (O.Holtz, B.Sturmfels, 2006) I Hyperdeterminants as discrete integrable systems: 2 × 2 and 2 × 2 × 2. I Is 2 × 2 × 2 × 2 hyperdeterminant integrable?? Plan: I The simplest (hyper)determinants: 2 × 2 and 2 × 2 × 2. I Modern applications of hyperdeterminants I The classical heritage: A.Cayley et al. Thomas Wolf Hyperdeterminants: a CA challenge I The next step: the (a?) 2 × 2 × 2 × 2 hyperdeterminant (B.Sturmfels et al., 2006) I FORM computations: how far can we reach now? I Principal Minor Assignment Problem (O.Holtz, B.Sturmfels, 2006) I Hyperdeterminants as discrete integrable systems: 2 × 2 and 2 × 2 × 2. I Is 2 × 2 × 2 × 2 hyperdeterminant integrable?? Plan: I The simplest (hyper)determinants: 2 × 2 and 2 × 2 × 2. I Modern applications of hyperdeterminants I The classical heritage: A.Cayley et al. I The definition of hyperdeterminants and its variations Thomas Wolf Hyperdeterminants: a CA challenge I FORM computations: how far can we reach now? I Principal Minor Assignment Problem (O.Holtz, B.Sturmfels, 2006) I Hyperdeterminants as discrete integrable systems: 2 × 2 and 2 × 2 × 2. I Is 2 × 2 × 2 × 2 hyperdeterminant integrable?? Plan: I The simplest (hyper)determinants: 2 × 2 and 2 × 2 × 2. I Modern applications of hyperdeterminants I The classical heritage: A.Cayley et al. I The definition of hyperdeterminants and its variations I The next step: the (a?) 2 × 2 × 2 × 2 hyperdeterminant (B.Sturmfels et al., 2006) Thomas Wolf Hyperdeterminants: a CA challenge I Principal Minor Assignment Problem (O.Holtz, B.Sturmfels, 2006) I Hyperdeterminants as discrete integrable systems: 2 × 2 and 2 × 2 × 2. I Is 2 × 2 × 2 × 2 hyperdeterminant integrable?? Plan: I The simplest (hyper)determinants: 2 × 2 and 2 × 2 × 2. I Modern applications of hyperdeterminants I The classical heritage: A.Cayley et al. I The definition of hyperdeterminants and its variations I The next step: the (a?) 2 × 2 × 2 × 2 hyperdeterminant (B.Sturmfels et al., 2006) I FORM computations: how far can we reach now? Thomas Wolf Hyperdeterminants: a CA challenge I Hyperdeterminants as discrete integrable systems: 2 × 2 and 2 × 2 × 2. I Is 2 × 2 × 2 × 2 hyperdeterminant integrable?? Plan: I The simplest (hyper)determinants: 2 × 2 and 2 × 2 × 2. I Modern applications of hyperdeterminants I The classical heritage: A.Cayley et al. I The definition of hyperdeterminants and its variations I The next step: the (a?) 2 × 2 × 2 × 2 hyperdeterminant (B.Sturmfels et al., 2006) I FORM computations: how far can we reach now? I Principal Minor Assignment Problem (O.Holtz, B.Sturmfels, 2006) Thomas Wolf Hyperdeterminants: a CA challenge I Is 2 × 2 × 2 × 2 hyperdeterminant integrable?? Plan: I The simplest (hyper)determinants: 2 × 2 and 2 × 2 × 2. I Modern applications of hyperdeterminants I The classical heritage: A.Cayley et al. I The definition of hyperdeterminants and its variations I The next step: the (a?) 2 × 2 × 2 × 2 hyperdeterminant (B.Sturmfels et al., 2006) I FORM computations: how far can we reach now? I Principal Minor Assignment Problem (O.Holtz, B.Sturmfels, 2006) I Hyperdeterminants as discrete integrable systems: 2 × 2 and 2 × 2 × 2. Thomas Wolf Hyperdeterminants: a CA challenge Plan: I The simplest (hyper)determinants: 2 × 2 and 2 × 2 × 2. I Modern applications of hyperdeterminants I The classical heritage: A.Cayley et al. I The definition of hyperdeterminants and its variations I The next step: the (a?) 2 × 2 × 2 × 2 hyperdeterminant (B.Sturmfels et al., 2006) I FORM computations: how far can we reach now? I Principal Minor Assignment Problem (O.Holtz, B.Sturmfels, 2006) I Hyperdeterminants as discrete integrable systems: 2 × 2 and 2 × 2 × 2. I Is 2 × 2 × 2 × 2 hyperdeterminant integrable?? Thomas Wolf Hyperdeterminants: a CA challenge I 2 × 2 × 2 a011 a111 a010 a110 a001 a101 a000 a100 hyperdet = 2 2 2 2 2 2 2 2 a111a000 +a100a011 +a101a010 +a110a001 − 2a111a110a001a000 − 2a111a101a010a000 − 2a111a100a011a000 − 2a110a101a010a001 − 2a110a100a011a001 − 2a101a100a011a010 + 4a111a100a010a001 + 4a110a101a011a000 2 × 2 and 2 × 2 × 2 (hyper)determinants I 2 × 2 a01 a11 a00 a10 det = a00a11 − a01a10 Thomas Wolf Hyperdeterminants: a CA challenge 2 × 2 and 2 × 2 × 2 (hyper)determinants I 2 × 2 × 2 a011 a111 I 2 × 2 a010 a110 a01 a11 a001 a101 a000 a100 a00 a10 hyperdet = det = a00a11 − a01a10 2 2 2 2 2 2 2 2 a111a000 +a100a011 +a101a010 +a110a001 − 2a111a110a001a000 − 2a111a101a010a000 − 2a111a100a011a000 − 2a110a101a010a001 − 2a110a100a011a001 − 2a101a100a011a010 + 4a111a100a010a001 + 4a110a101a011a000 Thomas Wolf Hyperdeterminants: a CA challenge Modern applications of hyperdeterminants I quantum information ("Multipartite Entanglement and Hyperdeterminants", A. Miyake, M. Wadati, Quant. Info. Comp. 2 (Special), 540-555 (2002), I biomathematics ("Estimating vaccine coverage by using computer algebra", R.Altmann, K. Altmann, Mathematical Medicine and Biology 2000 17(2):137-146, I numerical analysis ("Tensor rank and the ill-posedness of the best low-rank approximation problem", V. de Silva and L.-H. Lim, http://arxiv.org/abs/math/0607647) I data analysis ("Kruskal’s condition for uniqueness in Candecomp/Parafac..." A. Stegeman, J. M. F. Ten Berge Computational Statistics & Data Analysis 50(1): 210-220 (2006)), I a few other fields, in particular discrete integrable systems. Thomas Wolf Hyperdeterminants: a CA challenge The polynomial invariants of four qubits Jean-Gabriel Luque∗ and Jean-Yves Thibon† Institut Gaspard Monge, Universit´ede Marne-la-Vall´ee 77454 Marne-la-Vall´ee cedex, France (Dated: December 14, 2002) We describe explicitly the algebra of polynomial functions on the Hilbert space of four qubit states 4 which are invariant under the SLOCC group SL(2, C ) . From this description, we obtain a closed formula for the hyperdeterminant in terms of low degree invariants. PACS numbers: O3.67.Hk, 03.65.Ud, 03.65.Fd I. INTRODUCTION under the SLOCC group SL(2, C )4. This amounts to the construction of a moduli space for four qubit states. Our Various classifications of states with up to four qubits strategy is to find first the Hilbert series of the algebra have been recently proposed, with the aim of understand- of invariants J . Next, we construct by classical methods ing the different ways in which multipartite systems can four invariants of the required degrees. The knowledge be entangled [1, 2, 3, 4, 5]. However, one cannot ex- of the Hilbert series reduces then the proof of algebraic pect that such classifications will be worked out for an independence and completeness to simple verifications. arbitrary number k of qubits, and there is a need for a The values of the invariants on the orbits of [4] are tab- coarser classification scheme which would be computable ulated in the Appendix. for general k. In [6], Klyachko proposed to assimilate en- tanglement with the notion of semi-stability of geometric invariant theory. In this context, a semi-stable state is II. THE HILBERT SERIES one which can be separated from 0 by a polynomial in- C k variant of SL(2, ) , the point in the geometric approach Let Jd be the space of G-invariant homogeneous poly- being that explicit knowledge of the invariants is in prin- nomial functions of degree d in the variables Aijkl. Using ciple not necessary to check this property. some elementary representation theory, it is not difficult In this paper, we construct a complete set of algebraic to show that Jd is zero for d odd, and that for d = 2m invariants of 4-qubit states. This allows us to identify even, the dimension of Jd is equal to the multiplicity of arXiv:quant-ph/0212069v6 7 Jul 2005 PHYSICAL REVIEW D 73, 104033 (2006) Strings, black holes, and quantum information Renata Kallosh and Andrei Linde Department of Physics, Stanford University, Stanford, California 94305, USA (Received 22 February 2006; published 25 May 2006) We find multiple relations between extremal black holes in string theory and 2- and 3-qubit systems in quantum information theory. We show that the entropy of the axion-dilaton extremal black hole is related to the concurrence of a 2-qubit state, whereas the entropy of the STU black holes, Bogomol’nyi-Prasad- Sommerfield (BPS) as well as non-BPS, is related to the 3-tangle of a 3-qubit state.