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arXiv:quant-ph/0609227 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

E7 and the tripartite entanglement of seven qubits

M. J. Duff, S. Ferrara

Published Jul 19, 2007• 69 citations
arXivHTMLPDF

Abstract

In quantum information theory, it is well known that the tripartite entanglement of three qubits is described by the group $[SL(2,C){]}^{3}$ and that the entanglement measure is given by Cayley's hyperdeterminant. This has provided an analogy with certain $N=2$ supersymmetric black holes in string theory, whose entropy is also given by the hyperdeterminant. In this paper, we extend the analogy to $N=8$. We propose that a particular tripartite entanglement of seven qubits, encoded in the Fano plane, is described by the exceptional group ${E}_{7}(C)$ and that the entanglement measure is given by Cartan's quartic ${E}_{7}$ invariant.

DOI
10.1103/physrevd.76.025018

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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.
Recently it has been observed that the group E7{E}_{7}E7​ can be used to describe a special type of quantum entanglement of seven qubits partitioned into seven tripartite systems. Here we show that this curious type of entanglement is entirely encoded into the discrete geometry of the Fano plane. We explicitly work out the details concerning a qubit interpretation of the E7{E}_{7}E7​ generators as representatives of tripartite protocols acting on the 56-dimensional representation space. Using these results we extend further the recently studied analogy between quantum information theory and supersymmetric black holes in four-dimensional string theory. We point out that there is a dual relationship between entangled subsystems containing three and four tripartite systems. This relationship is reflected in the structure of the expressions for the black hole entropy in the N=4N=4N=4 and N=2N=2N=2 truncations of the E7(7){E}_{7(7)}E7(7)​ symmetric area form of N=8N=8N=8 supergravity. We conjecture that a similar picture based on other qubit systems might hold for black hole solutions in magic supergravities.
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,
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.
Twenty years ago, Cayley's hyperdeterminant, the degree four invariant of the polynomial ring \mathbb{C}[\mathbb{C}^2\otimes\mathbb{C}^2\otimes \mathbb{C}^2]^{{\operatorname{SL}_2(\mathbb{C})}^{\times 3}} , was popularized in modern physics as it separates genuine entanglement classes in the three-qubit Hilbert space and is connected to entropy formulas for special solutions of black holes. In this note, we compute the analogous invariant on the fermionic Fock space for N=8 , i.e., spin particles with four different locations, and show how this invariant projects to other well-known invariants in quantum information. We also give combinatorial interpretations of these formulas.
Recent work has established a correspondence between the tripartite entanglement measure of three qubits and the macroscopic entropy of the four-dimensional 8-charge STU black hole of supergravity. Here we consider the configurations of intersecting D3-branes, whose wrapping around the six compact dimensions T6 provides the microscopic string-theoretic interpretation of the charges, and associate the three-qubit basis vectors |ABC>, (A, B, C=0 or 1) with the corresponding 8 wrapping cycles. In particular, we relate a well-known fact of quantum information theory, that the most general real three-qubit state can be parameterized by four real numbers and an angle, to a well-known fact of string theory, that the most general STU black hole can be described by four D3-branes intersecting at an angle.