Abstract
We introduce “binding complexity”, a new notion of circuit complexity which quantifies the difficulty of distributing entanglement among multiple parties, each consisting of many local degrees of freedom. We define binding complexity of a given state as the minimal number of quantum gates that must act between parties to prepare it. To illustrate the new notion we compute it in a toy model for a scalar field theory, using certain multiparty entangled states which are analogous to configurations that are known in AdS/CFT to correspond to multiboundary wormholes. Pursuing this analogy, we show that our states can be prepared by the Euclidean path integral in (0 + 1)-dimensional quantum mechanics on graphs with wormhole-like structure. We compute the binding complexity of our states by adapting the Euler-Arnold approach to Nielsen’s geometrization of gate counting, and find a scaling with entropy that resembles a result for the interior volume of holographic multiboundary wormholes. We also compute the binding complexity of general coherent states in perturbation theory, and show that for “double-trace deformations” of the Hamiltonian the effects resemble expansion of a wormhole interior in holographic theories.
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P.W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM J. Comput. 26 (1997) 1484 [quant-ph/9508027].
W. Dür, G. Vidal and J.I. Cirac, Three qubits can be entangled in two inequivalent ways, Phys. Rev. A 62 (Nov, 2000) 062314 [quant-ph/0005115].
V. Coffman, J. Kundu and W.K. Wootters, Distributed entanglement, Phys. Rev. A 61 (2000) 052306 [quant-ph/9907047].
F. Diker, Deterministic construction of arbitrary w states with quadratically increasing number of two-qubit gates, arXiv:1606.09290.
R. Jefferson and R.C. Myers, Circuit complexity in quantum field theory, JHEP 10 (2017) 107 [arXiv:1707.08570] [INSPIRE].
M.A. Nielsen, A geometric approach to quantum circuit lower bounds, quant-ph/0502070.
K. Shizume, T. Nakajima, R. Nakayama and Y. Takahashi, Quantum computational Riemannian and sub-Riemannian geodesics, Prog. Theor. Phys. 127 (2012) 997 [INSPIRE].
M.R. Dowling and M.A. Nielsen, The geometry of quantum computation, quant-ph/0701004.
R. Khan, C. Krishnan and S. Sharma, Circuit complexity in fermionic field theory, Phys. Rev. D 98 (2018) 126001 [arXiv:1801.07620] [INSPIRE].
L. Hackl and R.C. Myers, Circuit complexity for free fermions, JHEP 07 (2018) 139 [arXiv:1803.10638] [INSPIRE].
M. Guo, J. Hernandez, R.C. Myers and S.-M. Ruan, Circuit complexity for coherent states, JHEP 10 (2018) 011 [arXiv:1807.07677] [INSPIRE].
A. Bhattacharyya, A. Shekar and A. Sinha, Circuit complexity in interacting QFTs and RG flows, JHEP 10 (2018) 140 [arXiv:1808.03105] [INSPIRE].
H.A. Camargo et al., Complexity as a novel probe of quantum quenches: universal scalings and purifications, arXiv:1807.07075 [INSPIRE].
D.W.F. Alves and G. Camilo, Evolution of complexity following a quantum quench in free field theory, JHEP 06 (2018) 029 [arXiv:1804.00107] [INSPIRE].
R.-Q. Yang and K.-Y. Kim, Complexity of operators generated by quantum mechanical Hamiltonians, arXiv:1810.09405 [INSPIRE].
R.-Q. Yang, Complexity for quantum field theory states and applications to thermofield double states, Phys. Rev. D 97 (2018) 066004 [arXiv:1709.00921] [INSPIRE].
R.-Q. Yang, C. Niu, C.-Y. Zhang and K.-Y. Kim, Comparison of holographic and field theoretic complexities for time dependent thermofield double states, JHEP 02 (2018) 082 [arXiv:1710.00600] [INSPIRE].
R.-Q. Yang et al., Principles and symmetries of complexity in quantum field theory, Eur. Phys. J. C 79 (2019) 109 [arXiv:1803.01797] [INSPIRE].
R.-Q. Yang et al., More on complexity of operators in quantum field theory, arXiv:1809.06678 [INSPIRE].
S. Chapman, M.P. Heller, H. Marrochio and F. Pastawski, Toward a definition of complexity for quantum field theory states, Phys. Rev. Lett. 120 (2018) 121602 [arXiv:1707.08582] [INSPIRE].
B. Czech, Einstein equations from varying complexity, Phys. Rev. Lett. 120 (2018) 031601 [arXiv:1706.00965] [INSPIRE].
P. Caputa et al., Anti-de Sitter space from optimization of path integrals in conformal field theories, Phys. Rev. Lett. 119 (2017) 071602 [arXiv:1703.00456] [INSPIRE].
P. Caputa et al., Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT, JHEP 11 (2017) 097 [arXiv:1706.07056] [INSPIRE].
A. Bhattacharyya et al., Path-integral complexity for perturbed CFTs, JHEP 07 (2018) 086 [arXiv:1804.01999] [INSPIRE].
T. Takayanagi, Holographic spacetimes as quantum circuits of path-integrations, JHEP 12 (2018) 048 [arXiv:1808.09072] [INSPIRE].
J. Molina-Vilaplana and A. Del Campo, Complexity functionals and complexity growth limits in continuous MERA circuits, JHEP 08 (2018) 012 [arXiv:1803.02356] [INSPIRE].
A. Belin, A. Lewkowycz and G. Sárosi, The boundary dual of the bulk symplectic form, Phys. Lett. B 789 (2019) 71 [arXiv:1806.10144] [INSPIRE].
A. Belin, A. Lewkowycz and G. Sárosi, Complexity and the bulk volume, a new York time story, arXiv:1811.03097 [INSPIRE].
J.M. Magán, Black holes, complexity and quantum chaos, JHEP 09 (2018) 043 [arXiv:1805.05839] [INSPIRE].
P. Caputa and J.M. Magan, Quantum computation as gravity, arXiv:1807.04422 [INSPIRE].
T. Ali et al., Time evolution of complexity: a critique of three methods, arXiv:1810.02734 [INSPIRE].
V. Arnold, Sur la géométrie différentielle des groupes de lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier 16 (1966) 319.
T. Tao, The Euler-Arnold equation, https://terrytao.wordpress.com/2010/06/07/the-euler-arnold-equation/ (2010).
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
V. Balasubramanian et al., Multiboundary wormholes and holographic entanglement, Class. Quant. Grav. 31 (2014) 185015 [arXiv:1406.2663] [INSPIRE].
D. Stanford and L. Susskind, Complexity and shock wave geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].
L. Susskind, Computational complexity and black hole horizons, Fortsch. Phys. 64 (2016) 44 [arXiv:1403.5695] [INSPIRE].
L. Susskind, Computational complexity and black hole horizons, Fortsch. Phys. 64 (2016) 44 [arXiv:1403.5695] [INSPIRE].
L. Susskind and Y. Zhao, Switchbacks and the bridge to nowhere, arXiv:1408.2823 [INSPIRE].
A.R. Brown and L. Susskind, Second law of quantum complexity, Phys. Rev. D 97 (2018) 086015 [arXiv:1701.01107] [INSPIRE].
D.R. Brill, Multi-black hole geometries in (2 + 1)-dimensional gravity, Phys. Rev. D 53 (1996) 4133 [gr-qc/9511022] [INSPIRE].
D. Brill, Black holes and wormholes in (2 + 1)-dimensions, gr-qc/9904083 [INSPIRE].
S. Aminneborg et al., Black holes and wormholes in (2 + 1)-dimensions, Class. Quant. Grav. 15 (1998) 627 [gr-qc/9707036] [INSPIRE].
K. Skenderis and B.C. van Rees, Holography and wormholes in 2 + 1 dimensions, Commun. Math. Phys. 301 (2011) 583 [arXiv:0912.2090] [INSPIRE].
K. Krasnov, Holography and Riemann surfaces, Adv. Theor. Math. Phys. 4 (2000) 929 [hep-th/0005106] [INSPIRE].
K. Krasnov, Black hole thermodynamics and Riemann surfaces, Class. Quant. Grav. 20 (2003) 2235 [gr-qc/0302073] [INSPIRE].
D. Marolf, H. Maxfield, A. Peach and S.F. Ross, Hot multiboundary wormholes from bipartite entanglement, Class. Quant. Grav. 32 (2015) 215006 [arXiv:1506.04128] [INSPIRE].
Z. Fu et al., Holographic complexity is nonlocal, JHEP 02 (2018) 072 [arXiv:1801.01137] [INSPIRE].
A. Peach and S.F. Ross, Tensor network models of multiboundary wormholes, Class. Quant. Grav. 34 (2017) 105011 [arXiv:1702.05984] [INSPIRE].
P. Gao, D.L. Jafferis and A. Wall, Traversable wormholes via a double trace deformation, JHEP 12 (2017) 151 [arXiv:1608.05687] [INSPIRE].
M.A. Nielsen et al., Quantum dynamics as a physical resource, Phys. Rev. A 67 (2003) 052301 [quant-ph/0208077].
A. Peres, Separability criterion for density matrices, Phys. Rev. Lett. 77 (1996) 1413 [quant-ph/9604005].
M. Horodecki, P. Horodecki and R. Horodecki, Separability of mixed states: necessary and sufficient conditions, Phys. Lett. A 223 (1996) 1 [quant-ph/9605038].
G. Vidal and R.F. Werner, Computable measure of entanglement, Phys. Rev. A 65 (2002) 032314 [quant-ph/0102117].
S. Rana, Negative eigenvalues of partial transposition of arbitrary bipartite states, Phys. Rev. A 87 (2013) 054301 [arXiv:1304.6775].
M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].
R. Simon, Peres-horodecki separability criterion for continuous variable systems, Phys. Rev. Lett. 84 (2000) 2726 [quant-ph/9909044].
R.F. Werner and M.M. Wolf, Bound entangled gaussian states, Phys. Rev. Lett. 86 (2001) 3658 [quant-ph/0009118].
E. Shchukin and W. Vogel, Inseparability criteria for continuous bipartite quantum states, Phys. Rev. Lett. 95 (2005) 230502 [quant-ph/0508132].
M. Bañados, C. Teitelboim and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].
A.R. Brown et al., Complexity, action and black holes, Phys. Rev. D 93 (2016) 086006 [arXiv:1512.04993] [INSPIRE].
P. Hosur, X.-L. Qi, D.A. Roberts and B. Yoshida, Chaos in quantum channels, JHEP 02 (2016) 004 [arXiv:1511.04021] [INSPIRE].
L. Susskind, L. Thorlacius and J. Uglum, The stretched horizon and black hole complementarity, Phys. Rev. D 48 (1993) 3743 [hep-th/9306069] [INSPIRE].
M. Freedman and M. Headrick, Bit threads and holographic entanglement, Commun. Math. Phys. 352 (2017) 407 [arXiv:1604.00354] [INSPIRE].
M. Headrick and V.E. Hubeny, Riemannian and Lorentzian flow-cut theorems, Class. Quant. Grav. 35 (2018) 10 [arXiv:1710.09516] [INSPIRE].
P. Hayden, M. Headrick and A. Maloney, Holographic mutual information is monogamous, Phys. Rev. D 87 (2013) 046003 [arXiv:1107.2940] [INSPIRE].
V.E. Hubeny, Bulk locality and cooperative flows, JHEP 12 (2018) 068 [arXiv:1808.05313] [INSPIRE].
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Balasubramanian, V., DeCross, M., Kar, A. et al. Binding complexity and multiparty entanglement. J. High Energ. Phys. 2019, 69 (2019). https://doi.org/10.1007/JHEP02(2019)069
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DOI: https://doi.org/10.1007/JHEP02(2019)069