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Fortschritte der Physik

E7(7) invariant measures of entanglement

L. Borsten

Published Jul 21, 2008• 24 citations
Publisher

Abstract

Abstract A study, centred on the Freudenthal triple system, of the E7(7) invariant tripartite entanglement of seven qubits and its relationship to black hole entropy in N = 8 supergravity. Based on work done in collaboration with D. Dahanayake, M. J. Duff and H. Ebrahim.
DOI
10.1002/prop.200810542

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In quantum information theory, it is well known that the tripartite entanglement of three qubits is described by the group [SL(2,C)]3[SL(2,C){]}^{3}[SL(2,C)]3 and that the entanglement measure is given by Cayley's hyperdeterminant. This has provided an analogy with certain N=2N=2N=2 supersymmetric black holes in string theory, whose entropy is also given by the hyperdeterminant. In this paper, we extend the analogy to N=8N=8N=8. We propose that a particular tripartite entanglement of seven qubits, encoded in the Fano plane, is described by the exceptional group E7(C){E}_{7}(C)E7​(C) and that the entanglement measure is given by Cartan's quartic E7{E}_{7}E7​ invariant.
We show that the 3-qubit entanglement classification is naturally encoded by that elegant algebraic structure, well known from supergravity, the Freudenthal triple system . In particular, we show that the six entanglement classes: (1) Separable A-B-C , (2a) Biseparable A-BC , (2b) Biseparable B-CA , (2c) Biseparable C-AB , (3) W and (4) Greenberger-Horne-Zeilinger correspond respectively to the Freudenthal triple ranks 1, 2a, 2b, 2c, 3, 4. Based on work done in collaboration with D. Dahanayake, M. J. Duff, H. Ebrahim and W. Rubens.
Recent investigations have established an analogy between the entropy of four-dimensional supersymmetric black holes in string theory and entanglement in quantum information theory. Examples include: (1) N=2N=2N=2 STU black holes and the tripartite entanglement of three qubits (2-state systems), where the common symmetry is [SL(2)]3[SL(2){]}^{3}[SL(2)]3 and (2) N=8N=8N=8 black holes and the tripartite entanglement of seven qubits where the common symmetry is ${E}_{7}\ensuremath{\supset}[SL(2){]}^{7}$. Here we present another example: N=8N=8N=8 black holes (or black strings) in five dimensions and the bipartite entanglement of three qutrits (3-state systems), where the common symmetry is ${E}_{6}\ensuremath{\supset}[SL(3){]}^{3}$. Both the black hole (or black string) entropy and the entanglement measure are provided by the Cartan cubic E6{E}_{6}E6​ invariant. Similar analogies exist for magic N=2N=2N=2 supergravity black holes in both four and five dimensions.
We review the recently established relationships between black hole entropy in string theory and the quantum entanglement of qubits and qutrits in quantum information theory. The first example is provided by the measure of the tripartite entanglement of three qubits (Alice, Bob and Charlie), known as the 3-tangle, and the entropy of the 8-charge ST U black hole of N = 2 supergravity, both of which are given by the [SL(2)] 3 invariant hyperdeterminant, a quantity first introduced by Cayley in 1845. Moreover the classification of three-qubit entanglements is related to the classification of N = 2 supersymmetric ST U black holes. There are further relationships between the attractor mechanism and local distillation protocols and between supersymmetry and the suppression of bit flip errors. At the microscopic level, the black holes are described by intersecting D3-branes whose wrapping around the six compact dimensions T 6 provides the string-theoretic interpretation of the charges and we associate the three-qubit basis vectors, |ABC〉 (A, B, C = 0 or 1), with the corresponding 8 wrapping cycles. The black hole/qubit correspondence extends to the 56 charge N = 8 black holes and the tripartite entanglement of seven qubits where the measure is provided by Cartan’s E7 ⊃ [SL(2)] 7 invariant. The qubits are naturally described by the seven vertices ABCDEF G of the Fano plane, which provides the multiplication table of the seven imaginary octonions,