Communications in Mathematical Physics

, Volume 342, Issue 1, pp 47–80

Product-State Approximations to Quantum States

Article

Abstract

We show that for any many-body quantum state there exists an unentangled quantum state such that most of the two-body reduced density matrices are close to those of the original state. This is a statement about the monogamy of entanglement, which cannot be shared without limit in the same way as classical correlation. Our main application is to Hamiltonians that are sums of two-body terms. For such Hamiltonians we show that there exist product states with energy that is close to the ground-state energy whenever the interaction graph of the Hamiltonian has high degree. This proves the validity of mean-field theory and gives an explicitly bounded approximation error. If we allow states that are entangled within small clusters of systems but product across clusters then good approximations exist when the Hamiltonian satisfies one or more of the following properties: (1) high degree, (2) small expansion, or (3) a ground state where the blocks in the partition have sublinear entanglement. Previously this was known only in the case of small expansion or in the regime where the entanglement was close to zero. Our approximations allow an extensive error in energy, which is the scale considered by the quantum PCP (probabilistically checkable proof) and NLTS (no low-energy trivial-state) conjectures. Thus our results put restrictions on the possible Hamiltonians that could be used for a possible proof of the qPCP or NLTS conjectures. By contrast the classical PCP constructions are often based on constraint graphs with high degree. Likewise we show that the parallel repetition that is possible with classical constraint satisfaction problems cannot also be possible for quantum Hamiltonians, unless qPCP is false. The main technical tool behind our results is a collection of new classical and quantum de Finetti theorems which do not make any symmetry assumptions on the underlying states.

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References

  1. 1.
    Aharonov, D., Arad, I., Landau, Z., Vazirani, U.: The detectability lemma and quantum gap amplification. In: STOC ’09, pp. 417–426 (2009). arXiv:0811.3412.
  2. 2.
    Aharonov D., Arad I., Vidick T.: Guest column: the quantum PCP conjecture. SIGACT News. 44(2), 47–49 (2013)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Aharonov, D., Eldar, L.: On the complexity of commuting local Hamiltonians, and tight conditions for topological order in such systems. In: FOCS ’11, pp. 334–343 (2011). arXiv:1102.0770
  4. 4.
    Aharonov, D., Eldar, L.: Commuting local Hamiltonians on expanders, locally testable quantum codes, and the qPCP conjecture (2013). arXiv:1301.3407
  5. 5.
    Aharonov D., Irani Gottesman S., Kempe J.: The power of quantum systems on a line. Commun. Math. Phys. 287(1), 41–65 (2009). arXiv:0705.4077 CrossRefADSMATHGoogle Scholar
  6. 6.
    Arora, S.: How np got a new definition: a survey of probabilistically checkable proofs. In: Proceedings of the ICM, vol. 3, pp 637–648, Beijing (2002). arXiv:cs/0304038
  7. 7.
    Arora S., Lund C., Motwani R., Sudan M., Szegedy M.: Proof verification and the hardness of approximation problems. JACM 45(3), 501–555 (1998)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Arora S., Safra S.: Probabilistic checking of proofs: a new characterization of NP. JACM 45(1), 70–122 (1998)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Barak, B., Raghavendra, P., Steurer, D.: Rounding semidefinite programming hierarchies via global correlation. In: FOCS ’11 (2011). arXiv:1104.4680
  10. 10.
    Bookatz A.D.: QMA-complete problems. Quantum Inf. Comput. 17(5), 5–6 (2014)MathSciNetGoogle Scholar
  11. 11.
    Brandao, F.G., Harrow, A.W.: Quantum de Finetti theorems under local measurements with applications. In: Proceedings of the 45th annual ACM symposium on Symposium on theory of computing, STOC ’13, pp. 861–870 (2013). arXiv:1210.6367
  12. 12.
    Brandão F.G.S.L., Christandl M., Yard J.: Faithful squashed entanglement. Commun. Math. Phys. 306(3), 805–830 (2011) arXiv:1010.1750 CrossRefADSMATHGoogle Scholar
  13. 13.
    Bravyi S., DiVincenzo D., Loss D., Terhal B.: Quantum simulation of many-body Hamiltonians using perturbation theory with bounded-strength interactions. Phys. Rev. Lett. 101(7), 70503 (2008) arXiv:0803.2686 CrossRefADSGoogle Scholar
  14. 14.
    Bravyi S., Vyalyi M.: Commutative version of the local Hamiltonian problem and common eigenspace problem. Quant. Inf. Comp. 5(3), 187–215 (2005) arXiv:quant-ph/0308021 MathSciNetMATHGoogle Scholar
  15. 15.
    Caves C.M., Fuchs C.A., Schack R.: Unknown quantum states: the quantum de Finetti representation. J. Math. Phys. 43(9), 4537–4559 (2002) arXiv:quant-ph/0104088 CrossRefADSMathSciNetMATHGoogle Scholar
  16. 16.
    Christandl M., König R., Mitchison G., Renner R.: One-and-a-half quantum de Finetti theorems. Commun. Math. Phys. 273, 473–498 (2007) arXiv:quant-ph/0602130 CrossRefADSMATHGoogle Scholar
  17. 17.
    Diaconis P., Freedman D.: Finite exchangeable sequences. Ann. Probab. 8, 745–764 (1980)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Dinur I.: The PCP theorem by gap amplification. J. ACM 54(3), 12 (2007)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Doherty, A.C., Wehner, S.: Convergence of SDP hierarchies for polynomial optimization on the hypersphere (2012). arXiv:1210.5048
  20. 20.
    Eisert J., Cramer M., Plenio M.: Area laws for the entanglement entropy: a review. Rev. Mod. Phys. 82(1), 277 (2010) arXiv:0808.3773 CrossRefADSMathSciNetMATHGoogle Scholar
  21. 21.
    Fannes M., Nachtergaele B., Werner R.: Finitely correlated states on quantum spin chains. Commun. Math. Phys. 144, 443–490 (1992)CrossRefADSMathSciNetMATHGoogle Scholar
  22. 22.
    Freedman M.H., Hastings M.B.: Quantum systems on non-k-hyperfinite complexes: a generalization of classical statistical mechanics on expander graphs. Quant. Inf.Compt. 14(1-2), 144–180 (2014)MathSciNetGoogle Scholar
  23. 23.
    Gharibian S., Kempe J.: Approximation algorithms for QMA-complete problems. SIAM J. Comput. 41(4), 1028–1050 (2012) arXiv:1101.3884 CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    Gharibian, S., Kempe, J., Regev, O.: Personal communication (2012)Google Scholar
  25. 25.
    Gottesman, D., Irani, S.: The quantum and classical complexity of translationally invariant tiling and Hamiltonian problems. In: FOCS ’09, pp. 95–104 (2009). arXiv:0905.2419
  26. 26.
    Harrow, A.W.: The church of the symmetric subspace (2013). arXiv:1308.6595
  27. 27.
    Hastings M.: Matrix product operators and central elements: classical description of a quantum state. Geometr. Topol. Monogr. 18, 115–160 (2012)CrossRefMathSciNetGoogle Scholar
  28. 28.
    Hastings M.B.: Trivial low energy states for commuting Hamiltonians, and the quantum PCP conjecture. Quant. Inf. Compt. 13(5-6), 393–429 (2013)MathSciNetGoogle Scholar
  29. 29.
    Hayden P., Jozsa R., Petz D., Winter A.: Structure of states which satisfy strong subadditivity of quantum entropy with equality. Commun. Math. Phys. 246, 359 (2004) arXiv:quant-ph/0304007 CrossRefADSMathSciNetMATHGoogle Scholar
  30. 30.
    Hoory S., Linial N., Wigderson A.: Expander graphs and their applications. Bull. Am. Math. Soc. 43(4), 439–562 (2006)CrossRefMathSciNetMATHGoogle Scholar
  31. 31.
    Horodecki R., Horodecki P., Horodecki M., Horodecki K.: Quantum entanglement. Rev. Mod. Phys. 81, 865–942 (2009) arXiv:quant-ph/0702225 CrossRefADSMathSciNetMATHGoogle Scholar
  32. 32.
    Hudson R.L., Moody G.R.: Locally normal symmetric states and an analogue of de Finetti’s theorem. Z. Wahrschein. verw. Geb. 33, 343–351 (1976)CrossRefMathSciNetMATHGoogle Scholar
  33. 33.
    Ibinson B., Linden N., Winter A.: Robustness of quantum Markov chains. Commun. Math. Phys. 277(2), 289–304 (2008) arXiv:quant-ph/0611057 CrossRefADSMathSciNetMATHGoogle Scholar
  34. 34.
    Kempe J., Kitaev A., Regev O.: The complexity of the local Hamiltonian problem. SIAM J. Comput. 35(5), 1070–1097 (2006)CrossRefMathSciNetMATHGoogle Scholar
  35. 35.
    Kitaev A.Y., Shen A.H., Vyalyi M.N.: Classical and quantum computation, volume 47 of Graduate Studies in Mathematics. AMS, Boston, MA (2002)Google Scholar
  36. 36.
    Koenig R., Renner R.: A de Finetti representation for finite symmetric quantum states. J. Math. Phys. 46(12), 122108 (2005) arXiv:quant-ph/0410229 CrossRefADSMathSciNetGoogle Scholar
  37. 37.
    Kraus B., Lewenstein M., Cirac J.: Ground states of fermionic lattice Hamiltonians with permutation symmetry. Phys. Rev. A. 88, 022335 (2013)CrossRefADSGoogle Scholar
  38. 38.
    Lancien C., Winter A.: Distinguishing multi-partite states by local measurements. Commun. Math.Phys. 323, 555–573 (2013)CrossRefADSMathSciNetMATHGoogle Scholar
  39. 39.
    Montanaro A.: Some applications of hypercontractive inequalities in quantum information theory. J. Math. Phys. 53(12), 122206 (2012) arXiv:1208.0161 CrossRefADSMathSciNetGoogle Scholar
  40. 40.
    Oliveira R., Terhal B.M.: The complexity of quantum spin systems on a two-dimensional square lattice. Quant. Inf. Comput. 8(10), 900–924 (2008) arXiv:quant-ph/0504050 MathSciNetMATHGoogle Scholar
  41. 41.
    Osborne T.: Hamiltonian complexity. Rep. Prog. Phys. 75(2), 022001 (2012)CrossRefADSMathSciNetGoogle Scholar
  42. 42.
    Raggio G.A., Werner R.F.: Quantum statistical mechanics of general mean field systems. Helv. Phys. Acta 62, 980–1003 (1989)MathSciNetMATHGoogle Scholar
  43. 43.
    Raghavendra, P., Tan, N.: Approximating CSPs with global cardinality constraints using SDP hierarchies. In: SODA ’12, pp. 373–387 (2012). arXiv:1110.1064
  44. 44.
    Raz R.: A parallel repetition theorem. SIAM J. Comput. 27(3), 763–803 (1998)CrossRefMathSciNetMATHGoogle Scholar
  45. 45.
    Renner R.: Symmetry implies independence. Nat. Phys. 3, 645–649 (2007) arXiv:quant-ph/0703069 CrossRefGoogle Scholar
  46. 46.
    Rodnianski I., Schlein B.: Quantum fluctuations and rate of convergence towards mean field dynamics. Commun. Math. Phys. 291, 31–61 (2007)CrossRefADSMathSciNetGoogle Scholar
  47. 47.
    Salavatipour, M.: Label cover, hardness of set cover (2011). http://webdocs.cs.ualberta.ca/~mreza/courses/Approx11/week12
  48. 48.
    Schuch N.: Complexity of commuting Hamiltonians on a square lattice of qubits. Quant. Inf. Comput. 11(11-12), 901–912 (2011) arXiv:1105.2843 MathSciNetMATHGoogle Scholar
  49. 49.
    Schuch N., Verstraete F.: Computational complexity of interacting electrons and fundamental limitations of density functional theory. Nat. Phys. 5(10), 732–735 (2009) arXiv:0712.0483 CrossRefGoogle Scholar
  50. 50.
    Schuch N., Wolf M., Verstraete F., Cirac J.: Entropy scaling and simulability by matrix product states. Phys. Rev. Lett. 100(3), 30504 (2008) arXiv:0705.0292 CrossRefADSMathSciNetGoogle Scholar
  51. 51.
    Størmer E.: Symmetric states of infinite tensor products of C *-algebras. J. Funct. Anal. 3, 48 (1969)CrossRefGoogle Scholar
  52. 52.
    Terhal B.: Is entanglement monogamous?. IBM J. Res. Dev. 48(1), 71–78 (2004)CrossRefGoogle Scholar
  53. 53.
    Trevisan, L.: Inapproximability of combinatorial optimization problems, Texcnical Report TR04-065, electronic collgium on computational complexity, (2004). arXiv:cs/0409043
  54. 54.
    Verstraete F., Cirac J.: Matrix product states represent ground states faithfully. Phys. Rev. B. 73(9), 094423 (2006) arXiv:cond-mat/0505140 CrossRefADSGoogle Scholar
  55. 55.
    Vidal G.: Efficient classical simulation of slightly entangled quantum computations. Phys. Rev. Lett. 91(14), 147902 (2003) arXiv:quant-ph/0301063 CrossRefADSGoogle Scholar
  56. 56.
    Watrous J.: Quantum computational complexity. In: Meyers, , R.A., (eds) Encyclopedia of complexity and system science., pp. 7174–8201. Springer, Berlin (2009)CrossRefGoogle Scholar
  57. 57.
    Werner R.F.: An application of Bell’s inequalities to a quantum state extension problem. Lett. Math. Phys. 17, 359 (1989)CrossRefADSMathSciNetMATHGoogle Scholar
  58. 58.
    Wolf M.M., Verstraete F., Hastings M.B., Cirac J.I.: Area laws in quantum systems: Mutual information and correlations. Phys. Rev. Lett. 100, 070502 (2008) arXiv:0704.3906 CrossRefADSMathSciNetGoogle Scholar
  59. 59.
    Yang D.: A simple proof of monogamy of entanglement. Phys. Lett. A 360(2), 249–250 (2006) arXiv:quant-ph/0604168 CrossRefADSMathSciNetMATHGoogle Scholar
  60. 60.
    Yang D., Horodecki K., Horodecki M., Horodecki P., Oppenheim J., Song W.: Squashed entanglement for multipartite states and entanglement measures based on the mixed convex roof. IEEE Trans. Inf. Theory 55(7), 3375–3387 (2009) arXiv:0704.2236 CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Fernando G. S. L. Brandão
    • 1
    • 2
  • Aram W. Harrow
    • 3
  1. 1.Quantum Architectures and Computation GroupMicrosoft ResearchRedmondUSA
  2. 2.Department of Computer ScienceUniversity College LondonLondonUK
  3. 3.Center for Theoretical PhysicsMassachusetts Institute of TechnologyCambridgeUSA

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