- Email: [email protected]
Black Holes and Entanglement
Nuclear Physics B (Proc. Suppl.) 216 (2011) 218–220 www.elsevier.com/locate/npbps
Black Holes and Entanglement L. Borstena a
Theoretical Physics, Blackett Laboratory, Imperial College London London SW7 2AZ, United Kingdom An unexpected interplay between the seemingly disparate fields of M-theory and Quantum Information has recently come to light. We summarise these developments, culminating in a classification of 4-qubit entanglement from the physics of ST U black holes. Based on work done in collaboration with D. Dahanayake, M. J. Duff, H. Ebrahim, A. Marrani and W. Rubens.
1. Introduction The irrefutable experimental successes of quantum theory demanded a radical reappraisal of our long held notions of “physical reality”. Quantum superposition reigns supreme; we reside in a fundamentally probabilistic universe. The philosophically comfortable tenets of local realism must be abandoned as phenomenologically insufficient - a direct consequence of that most quantum of phenomena, entanglement [1, 2]. It subsequently became apparent that, since information is created, stored, transformed and destroyed by physical processes, such a drastic reassessment of reality must have some profound consequences for our theories of information and computation. Quantum Information Theory (QIT) is the outcome [3]. It is the study of information processing systems which rely on the fundamental properties of quantum mechanics. It is anticipated that QIT may be harnessed to go beyond what is computationally achievable on any classical device. In this context entanglement is seen as a resource that may be created and consumed in the course of a quantum information theoretic protocol. It is essential to the emerging technologies of quantum cryptography, computation and communication. If entanglement is a resource one must address the question of its classification and quantification. It is crucial from both a technological and foundational perspective. The other pillar of XX-century physics, Einstein’s General Theory of Relativity, raises its own set of quandaries. Primarily, direct attempts 0920-5632/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2011.04.159
at quantizing gravity using perturbative quantum field theory are plagued by uncontrollable infinities. Yet, a complete description of fundamental physics requires a consistent theory of quantum gravity. M-theory, which arose out of pioneering work on superstrings and supergravity, is a compelling framework for a unified theory of the fundamental interactions including quantum gravity. However, it is fundamentally non-perturbative and consequently remains largely mysterious, offering up only remote corners of its full structure. Just as entanglement has been central to developments in QIT, black holes are an important window into the non-perturbative aspects of M-theory. For the most part these important endeavors in quantum information and gravity have led separate lives. However, this contribution centres on a curious and unexpected interplay between these seemingly disparate themes. It constitutes one corner of the black-hole/qubit correspondence: a relationship between the entanglement of qubits, the basic units of quantum information, and the entropy of black holes in M-theory [4]. 2. Black Holes and Qubits The entropy formula for the 8-charge (four electric plus four magnetic) ST U black hole [5–7], appearing in M-theory compactified to four dimensions, is given by the ‘hyperdeterminant’ [8], a quantity introduced by the mathematician Cayley in 1845 [9]. Remarkably, the hyperdeterminant also shows up in the 3-tangle τABC , which
L. Borsten / Nuclear Physics B (Proc. Suppl.) 216 (2011) 218–220
Figure 1. The classification of three qubits (left) exactly matches the classification of black holes from N wrapped branes (right). Only the GHZ state has a nonzero 3-tangle and only the N = 4 black hole has nonzero entropy. measures the degree of tripartite entanglement shared by three qubits (Alice, Bob and Charlie) [10, 11]. A qubit is a two-state (up/down) quantum system; so three qubits can have eight different states corresponding to the eight charges of the black hole. The black hole entropy SBH is related to the 3-tangle by [8], SBH =
π√ τABC . 2
It was subsequently realised that there is in fact a one-to-one correspondence between the classification of 3-qubit entanglement and the classification of extremal ST U black holes [4, 12]. Three qubits may be entangled in six physically distinct ways: (1) Separable A-B-C, (2a) Biseparable A-BC, (2b) Biseparable B-CA, (2c) Biseparable C-AB, (3) W and (4) GHZ (GreenbergerHorne-Zeilinger) [13, 14]. See Figure 1. Only W and GHZ have genuine three-way entanglement and only GHZ maximally violates local realism reflected in the fact that it has non-zero 3-tangle. By embedding the ST U model in type II string theory compactified on a 6-torus one obtains a microscopic interpretation of the blackhole/correspondence in terms the supersymmetric wrapping configurations of D-branes [4, 15]. Consider the type IIB theory. A subset of the black hole charges corresponds to D3-branes wrapping the internal space [16]. To relate this
219
picture to QIT we split the 6-torus into three two 2-tori, one for each qubit. We demand that each D3-brane wraps a single circle of each 2-torus. Denoting a wrapped circle by a cross and an unwrapped circle by a nought, we identify an up (down) qubit state with xo (ox). To wrap or not to wrap; that is the qubit. Using this dictionary one finds that the supersymmetric configurations of one, two, three and four intersecting D3-branes reproduces precisely the six 3-qubit entanglement classes. See Figure 1. A fifth effective charge may be introduced while preserving supersymmetry by intersecting the fourth brane at an angle [17, 18]. This relates 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 [19], to a well-known fact of string theory, that the most general ST U black hole can be described by four D3-branes intersecting at an angle [17, 18]. Further work [4, 15, 20–30] has led to a more complete dictionary translating a variety of phenomena in one language to those in the other. It seems that we are, as yet, only glimpsing the tip of an iceberg. For example, the much more difficult problem of classifying 4-qubit entanglement has recently been addressed by invoking the blackhole/qubit correspondence [31, 32]. This is of experimental significance as 4-qubit entanglement is now achievable in the laboratory [33,34]. The stationary ST U black hole solutions may be studied by performing a time-like reduction [30, 35–37]. The resulting classification of stationary black holes is related to 4-qubit entanglement via the Kostant-Sekiguchi theorem [38,39]. Our main result is that there are 31 entanglement families which reduce to nine (in agreement with [40, 41]) up to permutations of the four qubits. From the black hole perspective, we find that the attractor equations, which determine the amount of supersymmetry preserved by a particular black hole solution, display a symmetry consistent with permutations of the qubits. For example, the A-GHZ state yields a set of attractor equations which are invariant under a triality corresponding to the permutation of B, C, D in the GHZ state. I would like to thank the organisers for a wonderful school in a beautiful setting.
220
L. Borsten / Nuclear Physics B (Proc. Suppl.) 216 (2011) 218–220
REFERENCES 1. Einstein, A., Podolsky, B., and Rosen, N. Phys. Rev. 47(10), 777–780 (1935). 2. Bell, J. S. Physics 1 (1964) no. 3, 195. 3. Nielsen, M. A. and Chuang, I. L. Quantum Computation and Quantum Information. Cambridge University Press, New York, NY, USA, (2000). 4. Borsten, L., Dahanayake, D., Duff, M. J., Ebrahim, H., and Rubens, W. Phys. Rep. 471(3–4), 113–219 (2009). 5. Duff, M. J., Liu, J. T., and Rahmfeld, J. Nucl. Phys. B459, 125–159 (1996). 6. Behrndt, K., Kallosh, R., Rahmfeld, J., Shmakova, M., and Wong, W. K. Phys. Rev. D54(10), 6293–6301 (1996). 7. Bellucci, S., Ferrara, S., Marrani, A., and Yeranyan, A. Entropy 10(4), 507–555 (2008). 8. Duff, M. J. Phys. Rev. D76, 025017 (2007). 9. Cayley, A. Camb. Math. J. 4 (1845) 193–209. 10. Coffman, V., Kundu, J., and Wootters, W. K. Phys. Rev. A61(5), 052306 (2000). 11. Miyake, A. Phys. Rev. A67(1), 012108 (2003). 12. Kallosh, R. and Linde, A. Phys. Rev. D73(10), 104033 (2006). 13. D¨ ur, W., Vidal, G., and Cirac, J. I. Phys. Rev. A62(6), 062314 (2000). 14. Borsten, L., Dahanayake, D., Duff, M. J., Rubens, W., and Ebrahim, H. Phys. Rev. A80, 032326 (2009). 15. Borsten, L., Dahanayake, D., Duff, M. J., Rubens, W., and Ebrahim, H. Phys. Rev. Lett. 100(25), 251602 (2008). 16. Klebanov, I. R. and Tseytlin, A. A. Nucl. Phys. B475, 179–192 (1996). 17. Balasubramanian, V. In Cargese 1997, Strings, branes and dualities, 399–410, (1997). Published in the proceedings of NATO Advanced Study Institute on Strings, Branes and Dualities, Cargese, France, 26 May - 14 Jun 1997. 18. Bertolini, M. and Trigiante, M. Nucl. Phys. B582, 393–406 (2000). 19. Acin, A., Andrianov, A., Jane, E., and Tarrach, R. J. Phys. A34(35), 6725–6739 (2001). 20. L´evay, P. Phys. Rev. D74(2), 024030 (2006).
21. Duff, M. J. and Ferrara, S. Phys. Rev. D76(2), 025018 (2007). 22. L´evay, P. Phys. Rev. D75(2), 024024 (2007). 23. Duff, M. J. and Ferrara, S. Phys. Rev. D76(12), 124023 (2007). 24. L´evay, P. Phys. Rev. D76(10), 106011 (2007). 25. Borsten, L. Fortschr. Phys. 56(7–9), 842–848 (2008). 26. L´evay, P., Saniga, M., and Vrana, P. Phys. Rev. D78(12), 124022 (2008). 27. Levay, P., Saniga, M., Vrana, P., and Pracna, P. Phys. Rev. D79, 084036 (2009). 28. Borsten, L., Dahanayake, D., Duff, M. J., and Rubens, W. Phys. Rev. D81, 105023 (2010). 29. Levay, P. and Szalay, S. (2010). 30. Levay, P. (2010). 31. Borsten, L., Dahanayake, D., Duff, M. J., Marrani, A., and Rubens, W. Phys. Rev. Lett. 105, 100507 (2010). 32. Borsten, L., Duff, M., Marrani, A., and Rubens, W. (2011). * Temporary entry *. 33. Amselem, E. and Bourennane, M. Nat. Phys. 5, 748–752 (2009). 34. Lavoie, J., Kaltenbaek, R., Piani, M., and Resch, K. J. Phys. Rev. Lett. 105(13), 130501 Sep (2010). 35. Breitenlohner, P., Maison, D., and Gibbons, G. W. Commun. Math. Phys. 120, 295 (1988). 36. Bergshoeff, E., Chemissany, W., Ploegh, A., Trigiante, M., and Van Riet, T. Nucl. Phys. B812, 343–401 (2009). 37. Bossard, G., Michel, Y., and Pioline, B. JHEP 01, 038 (2010). 38. Sekiguchi, J. J. Math. Soc. Japan 39(1), 127– 138 (1987). 39. Collingwood, D. H. and McGovern, W. M. Nilpotent orbits in semisimple Lie algebras. Van Nostrand Reinhold mathematics series. CRC Press, (1993). 40. Verstraete, F., Dehaene, J., De Moor, B., and Verschelde, H. Phys. Rev. A65(5), 052112 (2002). ˇ In Lin41. Chterental, O. and Djokovi´c, D. Z. ear Algebra Research Advances, Ling, G. D., editor, chapter 4, 133–167. Nova Science Publishers Inc (2007).
Black Holes and Entanglement L. Borstena a
Theoretical Physics, Blackett Laboratory, Imperial College London London SW7 2AZ, United Kingdom An unexpected interplay between the seemingly disparate fields of M-theory and Quantum Information has recently come to light. We summarise these developments, culminating in a classification of 4-qubit entanglement from the physics of ST U black holes. Based on work done in collaboration with D. Dahanayake, M. J. Duff, H. Ebrahim, A. Marrani and W. Rubens.
1. Introduction The irrefutable experimental successes of quantum theory demanded a radical reappraisal of our long held notions of “physical reality”. Quantum superposition reigns supreme; we reside in a fundamentally probabilistic universe. The philosophically comfortable tenets of local realism must be abandoned as phenomenologically insufficient - a direct consequence of that most quantum of phenomena, entanglement [1, 2]. It subsequently became apparent that, since information is created, stored, transformed and destroyed by physical processes, such a drastic reassessment of reality must have some profound consequences for our theories of information and computation. Quantum Information Theory (QIT) is the outcome [3]. It is the study of information processing systems which rely on the fundamental properties of quantum mechanics. It is anticipated that QIT may be harnessed to go beyond what is computationally achievable on any classical device. In this context entanglement is seen as a resource that may be created and consumed in the course of a quantum information theoretic protocol. It is essential to the emerging technologies of quantum cryptography, computation and communication. If entanglement is a resource one must address the question of its classification and quantification. It is crucial from both a technological and foundational perspective. The other pillar of XX-century physics, Einstein’s General Theory of Relativity, raises its own set of quandaries. Primarily, direct attempts 0920-5632/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2011.04.159
at quantizing gravity using perturbative quantum field theory are plagued by uncontrollable infinities. Yet, a complete description of fundamental physics requires a consistent theory of quantum gravity. M-theory, which arose out of pioneering work on superstrings and supergravity, is a compelling framework for a unified theory of the fundamental interactions including quantum gravity. However, it is fundamentally non-perturbative and consequently remains largely mysterious, offering up only remote corners of its full structure. Just as entanglement has been central to developments in QIT, black holes are an important window into the non-perturbative aspects of M-theory. For the most part these important endeavors in quantum information and gravity have led separate lives. However, this contribution centres on a curious and unexpected interplay between these seemingly disparate themes. It constitutes one corner of the black-hole/qubit correspondence: a relationship between the entanglement of qubits, the basic units of quantum information, and the entropy of black holes in M-theory [4]. 2. Black Holes and Qubits The entropy formula for the 8-charge (four electric plus four magnetic) ST U black hole [5–7], appearing in M-theory compactified to four dimensions, is given by the ‘hyperdeterminant’ [8], a quantity introduced by the mathematician Cayley in 1845 [9]. Remarkably, the hyperdeterminant also shows up in the 3-tangle τABC , which
L. Borsten / Nuclear Physics B (Proc. Suppl.) 216 (2011) 218–220
Figure 1. The classification of three qubits (left) exactly matches the classification of black holes from N wrapped branes (right). Only the GHZ state has a nonzero 3-tangle and only the N = 4 black hole has nonzero entropy. measures the degree of tripartite entanglement shared by three qubits (Alice, Bob and Charlie) [10, 11]. A qubit is a two-state (up/down) quantum system; so three qubits can have eight different states corresponding to the eight charges of the black hole. The black hole entropy SBH is related to the 3-tangle by [8], SBH =
π√ τABC . 2
It was subsequently realised that there is in fact a one-to-one correspondence between the classification of 3-qubit entanglement and the classification of extremal ST U black holes [4, 12]. Three qubits may be entangled in six physically distinct ways: (1) Separable A-B-C, (2a) Biseparable A-BC, (2b) Biseparable B-CA, (2c) Biseparable C-AB, (3) W and (4) GHZ (GreenbergerHorne-Zeilinger) [13, 14]. See Figure 1. Only W and GHZ have genuine three-way entanglement and only GHZ maximally violates local realism reflected in the fact that it has non-zero 3-tangle. By embedding the ST U model in type II string theory compactified on a 6-torus one obtains a microscopic interpretation of the blackhole/correspondence in terms the supersymmetric wrapping configurations of D-branes [4, 15]. Consider the type IIB theory. A subset of the black hole charges corresponds to D3-branes wrapping the internal space [16]. To relate this
219
picture to QIT we split the 6-torus into three two 2-tori, one for each qubit. We demand that each D3-brane wraps a single circle of each 2-torus. Denoting a wrapped circle by a cross and an unwrapped circle by a nought, we identify an up (down) qubit state with xo (ox). To wrap or not to wrap; that is the qubit. Using this dictionary one finds that the supersymmetric configurations of one, two, three and four intersecting D3-branes reproduces precisely the six 3-qubit entanglement classes. See Figure 1. A fifth effective charge may be introduced while preserving supersymmetry by intersecting the fourth brane at an angle [17, 18]. This relates 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 [19], to a well-known fact of string theory, that the most general ST U black hole can be described by four D3-branes intersecting at an angle [17, 18]. Further work [4, 15, 20–30] has led to a more complete dictionary translating a variety of phenomena in one language to those in the other. It seems that we are, as yet, only glimpsing the tip of an iceberg. For example, the much more difficult problem of classifying 4-qubit entanglement has recently been addressed by invoking the blackhole/qubit correspondence [31, 32]. This is of experimental significance as 4-qubit entanglement is now achievable in the laboratory [33,34]. The stationary ST U black hole solutions may be studied by performing a time-like reduction [30, 35–37]. The resulting classification of stationary black holes is related to 4-qubit entanglement via the Kostant-Sekiguchi theorem [38,39]. Our main result is that there are 31 entanglement families which reduce to nine (in agreement with [40, 41]) up to permutations of the four qubits. From the black hole perspective, we find that the attractor equations, which determine the amount of supersymmetry preserved by a particular black hole solution, display a symmetry consistent with permutations of the qubits. For example, the A-GHZ state yields a set of attractor equations which are invariant under a triality corresponding to the permutation of B, C, D in the GHZ state. I would like to thank the organisers for a wonderful school in a beautiful setting.
220
L. Borsten / Nuclear Physics B (Proc. Suppl.) 216 (2011) 218–220
REFERENCES 1. Einstein, A., Podolsky, B., and Rosen, N. Phys. Rev. 47(10), 777–780 (1935). 2. Bell, J. S. Physics 1 (1964) no. 3, 195. 3. Nielsen, M. A. and Chuang, I. L. Quantum Computation and Quantum Information. Cambridge University Press, New York, NY, USA, (2000). 4. Borsten, L., Dahanayake, D., Duff, M. J., Ebrahim, H., and Rubens, W. Phys. Rep. 471(3–4), 113–219 (2009). 5. Duff, M. J., Liu, J. T., and Rahmfeld, J. Nucl. Phys. B459, 125–159 (1996). 6. Behrndt, K., Kallosh, R., Rahmfeld, J., Shmakova, M., and Wong, W. K. Phys. Rev. D54(10), 6293–6301 (1996). 7. Bellucci, S., Ferrara, S., Marrani, A., and Yeranyan, A. Entropy 10(4), 507–555 (2008). 8. Duff, M. J. Phys. Rev. D76, 025017 (2007). 9. Cayley, A. Camb. Math. J. 4 (1845) 193–209. 10. Coffman, V., Kundu, J., and Wootters, W. K. Phys. Rev. A61(5), 052306 (2000). 11. Miyake, A. Phys. Rev. A67(1), 012108 (2003). 12. Kallosh, R. and Linde, A. Phys. Rev. D73(10), 104033 (2006). 13. D¨ ur, W., Vidal, G., and Cirac, J. I. Phys. Rev. A62(6), 062314 (2000). 14. Borsten, L., Dahanayake, D., Duff, M. J., Rubens, W., and Ebrahim, H. Phys. Rev. A80, 032326 (2009). 15. Borsten, L., Dahanayake, D., Duff, M. J., Rubens, W., and Ebrahim, H. Phys. Rev. Lett. 100(25), 251602 (2008). 16. Klebanov, I. R. and Tseytlin, A. A. Nucl. Phys. B475, 179–192 (1996). 17. Balasubramanian, V. In Cargese 1997, Strings, branes and dualities, 399–410, (1997). Published in the proceedings of NATO Advanced Study Institute on Strings, Branes and Dualities, Cargese, France, 26 May - 14 Jun 1997. 18. Bertolini, M. and Trigiante, M. Nucl. Phys. B582, 393–406 (2000). 19. Acin, A., Andrianov, A., Jane, E., and Tarrach, R. J. Phys. A34(35), 6725–6739 (2001). 20. L´evay, P. Phys. Rev. D74(2), 024030 (2006).
21. Duff, M. J. and Ferrara, S. Phys. Rev. D76(2), 025018 (2007). 22. L´evay, P. Phys. Rev. D75(2), 024024 (2007). 23. Duff, M. J. and Ferrara, S. Phys. Rev. D76(12), 124023 (2007). 24. L´evay, P. Phys. Rev. D76(10), 106011 (2007). 25. Borsten, L. Fortschr. Phys. 56(7–9), 842–848 (2008). 26. L´evay, P., Saniga, M., and Vrana, P. Phys. Rev. D78(12), 124022 (2008). 27. Levay, P., Saniga, M., Vrana, P., and Pracna, P. Phys. Rev. D79, 084036 (2009). 28. Borsten, L., Dahanayake, D., Duff, M. J., and Rubens, W. Phys. Rev. D81, 105023 (2010). 29. Levay, P. and Szalay, S. (2010). 30. Levay, P. (2010). 31. Borsten, L., Dahanayake, D., Duff, M. J., Marrani, A., and Rubens, W. Phys. Rev. Lett. 105, 100507 (2010). 32. Borsten, L., Duff, M., Marrani, A., and Rubens, W. (2011). * Temporary entry *. 33. Amselem, E. and Bourennane, M. Nat. Phys. 5, 748–752 (2009). 34. Lavoie, J., Kaltenbaek, R., Piani, M., and Resch, K. J. Phys. Rev. Lett. 105(13), 130501 Sep (2010). 35. Breitenlohner, P., Maison, D., and Gibbons, G. W. Commun. Math. Phys. 120, 295 (1988). 36. Bergshoeff, E., Chemissany, W., Ploegh, A., Trigiante, M., and Van Riet, T. Nucl. Phys. B812, 343–401 (2009). 37. Bossard, G., Michel, Y., and Pioline, B. JHEP 01, 038 (2010). 38. Sekiguchi, J. J. Math. Soc. Japan 39(1), 127– 138 (1987). 39. Collingwood, D. H. and McGovern, W. M. Nilpotent orbits in semisimple Lie algebras. Van Nostrand Reinhold mathematics series. CRC Press, (1993). 40. Verstraete, F., Dehaene, J., De Moor, B., and Verschelde, H. Phys. Rev. A65(5), 052112 (2002). ˇ In Lin41. Chterental, O. and Djokovi´c, D. Z. ear Algebra Research Advances, Ling, G. D., editor, chapter 4, 133–167. Nova Science Publishers Inc (2007).