Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
144 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Finding maximal quantum resources (2210.13475v3)

Published 24 Oct 2022 in quant-ph, cs.NA, and math.NA

Abstract: For many applications the presence of a quantum advantage crucially depends on the availability of resourceful states. Although the resource typically depends on the particular task, in the context of multipartite systems entangled quantum states are often regarded as resourceful. We propose an algorithmic method to find maximally resourceful states of several particles for various applications and quantifiers. We discuss in detail the case of the geometric measure, identifying physically interesting states and delivering insights to the problem of absolutely maximally entangled states. Moreover, we demonstrate the universality of our approach by applying it to maximally entangled subspaces, the Schmidt-rank, the stabilizer rank as well as the preparability in triangle networks.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (101)
  1. C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” Theor. Comput. Sci. 560, 7–11 (2014).
  2. A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661 (1991).
  3. M. J. Holland and K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71, 1355 (1993).
  4. X. Zhou, D. W. Leung,  and I. L. Chuang, “Methodology for quantum logic gate construction,” Phys. Rev. A 62, 052316 (2000).
  5. S. Bravyi and A. Kitaev, “Universal quantum computation with ideal Clifford gates and noisy ancillas,” Phys. Rev. Lett. 71, 022316 (2005).
  6. R. Raussendorf and H.J. Briegel, “A One-Way Quantum Computer,” Phys. Rev. Lett. 86, 5188 (2001).
  7. R. Raussendorf, D. E. Browne,  and H. J. Briegel, “Measurement-based quantum computation on cluster states,” Phys. Rev. A 68, 022312 (2003).
  8. A. J. Scott, “Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions,” Phys. Rev. A 69, 052330 (2004).
  9. P. Facchi, G. Florio, G. Parisi,  and S. Pascazio, “Maximally multipartite entangled states,” Phys. Rev. A 77, 060304(R) (2008).
  10. R. Reuvers, ‘‘An algorithm to explore entanglement in small systems,” Proc. R. Soc. A 474 (2018), 10.1098/rspa.2018.0023.
  11. G. Gour and N. R. Wallach, “All maximally entangled four-qubit states,” J. Math. Phys. 51, 112201 (2010).
  12. F. Huber, O. Gühne,  and J. Siewert, “Absolutely maximally entangled states of seven qubits do not exist,” Phys. Rev. Lett. 118, 200502 (2017).
  13. F. Huber, C. Eltschka, J. Siewert,  and O. Gühne, “Bounds on absolutely maximally entangled states from shadow inequalities, and the quantum MacWilliams identity,” J. Phys. A 51, 175301 (2018).
  14. D. Uzcátegui Contreras and D. Goyeneche, “Reconstructing the whole from its parts,” arXiv:2209.14154  (2022).
  15. P. Horodecki, Ł. Rudnicki,  and K. Życzkowski, “Five open problems in quantum information theory,” PRX Quantum 3, 010101 (2022).
  16. S. A. Rather, A. Burchardt, W. Bruzda, G. Rajchel-Mieldzioć, A. Lakshminarayan,  and K. Życzkowski, “Thirty-six entangled officers of Euler,” Phys. Rev. Lett. 128, 080507 (2022).
  17. V. Vedral and M. B. Plenio, “Entanglement measures and purification procedures,” Phys. Rev. A 57, 1619 (1998).
  18. G. Vidal and R. Tarrach, “Robustness of entanglement,” Phys. Rev. A 59, 141 (1998).
  19. O. Giraud, P. Braun,  and D. Braun, “Quantifying quantumness and the quest for queens of quantum,” New J. Phys. 12, 063005 (2010).
  20. A. Shimony, “Degree of Entanglement,” Ann. N. Y. Acad. Sci. 755, 675–679 (1995).
  21. H. Barnum and N. Linden, “Monotones and invariants for multi-particle quantum states,” J. Phys. A 34, 6787 (2001).
  22. T.-C. Wei and P. M. Goldbart, “Geometric measure of entanglement and applications to bipartite and multipartite quantum states,” Phys. Rev. A 68, 042307 (2003).
  23. P. Badzia̧g, Č. Brukner, W. Laskowski, T. Paterek,  and M. Z˙˙Z\dot{\text{Z}}over˙ start_ARG Z end_ARGukowski, “Experimentally Friendly Geometrical Criteria for Entanglement,” Phys. Rev. Lett. 100, 140403 (2008).
  24. M. Hayashi, D. Markham, M. Murao, M. Owari,  and S. Virmani, “Bounds on multipartite entangled orthogonal state discrimination using local operations and classical communication,” Phys. Rev. Lett. 96, 040501 (2006).
  25. A. Sen(De) and U. Sen, “Channel capacities versus entanglement measures in multiparty quantum states,” Phys. Rev. A 81, 012308 (2010).
  26. L. Chen, H. Zhu,  and T.-C. Wei, “Connections of geometric measure of entanglement of pure symmetric states to quantum state estimation,” Phys. Rev. A 83, 012305 (2011).
  27. R. Orús and T.-C. Wei, “Visualizing elusive phase transitions with geometric entanglement,” Phys. Rev. B 82, 155120 (2010).
  28. O. Buerschaper, A. García-Saez, R. Orús,  and T.-C. Wei, “Topological minimally entangled states via geometric measure,” J. Stat. Mech. , 11009 (2014).
  29. Q.-Q. Shi, H.-L. Wang, S.-H. Li, S. Y. Cho, M. T. Batchelor,  and H.-Q. Zhou, “Geometric entanglement and quantum phase transitions in two-dimensional quantum lattice models,” Phys. Rev. A 93, 062341 (2016).
  30. A. Deger and T.-C. Wei, “Geometric entanglement and quantum phase transition in generalized cluster-xy models,” Quantum Inf. Process. 18, 326 (2019).
  31. S. Gharibian and J. Kempe, “Approximation algorithms for QMA-complete problems,” SIAM J. Comput. 41, 1028–1050 (2012).
  32. A. Montanaro, “Injective tensor norms and open problems in quantum information,” https://people.maths.bris.ac.uk/ csxam/presentations /injnormtalk.pdf (2012).
  33. M. van den Nest, W. Dür, A. Miyake,  and H. J. Briegel, “Fundamentals of universality in one-way quantum computation,” New J. Phys. 9, 1–51 (2007).
  34. M. J. Bremner, C. Mora,  and A. Winter, “Are random pure states useful for quantum computation?” Phys. Rev. Lett. 102, 190502 (2009).
  35. D. Gross, S. T. Flammia,  and J. Eisert, “Most quantum states are too entangled to be useful as computational resources,” Phys. Rev. Lett. 102, 190501 (2009).
  36. M. Aulbach, D. Markham,  and M. Murao, “The maximally entangled symmetric state in terms of the geometric measure,” New J. Phys. 12, 073025 (2010).
  37. J. Martin, O. Giraud, P. A. Braun, D. Braun,  and T. Bastin, “Multiqubit symmetric states with high geometric entanglement,” Phys. Rev. A 81, 062347 (2010).
  38. M. Hajdušek. and M. Murao, “Direct evaluation of pure graph state entanglement,” New J. Phys. 15, 013039 (2013).
  39. E. Chitambar, R. Duan,  and Y. Shi, “Tripartite entanglement transformations and tensor rank,” Phys. Rev. Lett. 101, 140502 (2008).
  40. C. J. Hillar and L. H. Lim, “Most tensor problems are NP-hard,” J. ACM 60, 1–39 (2013).
  41. L. Qi, “Eigenvalues of a real supersymmetric tensor,” J. Symb. Comput. 40, 1302–1324 (2005).
  42. L. Chen, A. Xu,  and H. Zhu, “Computation of the geometric measure of entanglement for pure multiqubit states,” Phys. Rev. A 82, 032301 (2010).
  43. G. Ni, L. Qi,  and M. Bai, “Geometric measure of entanglement and U-eigenvalues of tensors,” SIAM J. Matrix Anal. Appl. 35, 73–87 (2014).
  44. S. Hu., L. Qi,  and G. Zhang, “Computing the geometric measure of entanglement of multipartite pure states by means of non-negative tensors,” Phys. Rev. A 93, 012304 (2016).
  45. A. Czaplinski, T. Raasch,  and J. Steinberg, “Real eigenstructure of regular simplex tensors,” Adv. Appl. Math. 148, 102521 (2023).
  46. R. F. Werner and A. S. Holevo, “Counterexample to an additivity conjecture for output purity of quantum channels,” J. Math. Phys. 43, 4353 (2002).
  47. G. Aubrun and S. J. Szarek, “Alice and Bob Meet Banach: The Interface of Asymptotic Geometric Analysis and Quantum Information Theory,”  (American Mathematical Society, 2017) Chap. 8.
  48. T.-C. Wei and S. Severini, “Matrix permanent and quantum entanglement of permutation invariant states,” J. Math. Phys. 51, 092203 (2010).
  49. V. de Silva and L.-H. Lim, “Tensor rank and the ill-posedness of the best low-rank approximation problem,” SIAM J. Matrix Anal. Appl. 30, 1084–1127 (2008).
  50. O. Gühne, M. Reimpell,  and R. F. Werner, “Estimating entanglement measures in experiments,” Phys. Rev. Lett. 98, 110502 (2007).
  51. A. Streltsov, H. Kampermann,  and D. Bruss, “Simple algorithm for computing the geometric measure of entanglement,” Phys. Rev. A 84, 022323 (2011).
  52. S. Gerke, W. Vogel,  and J. Sperling, “Numerical construction of multipartite entanglement witnesses,” Phys. Rev. X 8, 031047 (2018).
  53. Y. Nesterov, “Lectures on convex optimization,”  (Springer, 2018) Chap. Nonlinear Optimization.
  54. P. Mehta, M. Bukov, C.-H. Wang, A.G.R. Day, C. Richardson, C.K. Fisher,  and D.J. Schwab, “A high-bias, low-variance introduction to Machine Learning for physicists,” Phys. Rep. 810, 1–124 (2019).
  55. Y. Nesterov, “A method for unconstrained convex minimization problem with the rate of convergence o⁢(1/k2)𝑜1superscript𝑘2o(1/k^{2})italic_o ( 1 / italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ),” Soviet. Math. Docl. 269, 543–547 (1983).
  56. W. Dür, G. Vidal,  and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
  57. S. Tamaryan, T.-C. Wei,  and D. Park, “Maximally entangled three-qubit states via geometric measure of entanglement,” Phys. Rev. A 80, 052315 (2009).
  58. P. Krammer, H. Kampermann, D. Bruß, R. A. Bertlmann, L. C. Kwek,  and C. Macchiavello, “Multipartite entanglement detection via structure factors,” Phys. Rev. Lett. 103, 100502 (2009).
  59. A. Higuchi and A. Sudbery, ‘‘How entangled can two couples get?” Phys. Lett. A 273, 213–217 (2000).
  60. E. Knill, R. Laflamme, R. Martinez,  and C. Negrevergne, “Benchmarking quantum computers: The five-qubit error correcting code,” Phys. Rev. Lett. 86, 5811 (2001).
  61. N. Gisin and H. Bechmann-Pasquinucci, “Bell inequality, Bell states and maximally entangled states for n𝑛nitalic_n qubits,” Phys. Lett. A 246, 1–6 (1998).
  62. G. Vidal, “Entanglement monotones,” J. Mod. Opt. 47, 355–376 (2000).
  63. Z. Raissi, A. Teixidó, C. Gogolin,  and A. Acín, “Constructions of k-uniform and absolutely maximally entangled states beyond maximum distance codes,” Phys. Rev. Research 2, 033411 (2020).
  64. A. Burchardt and Z. Raissi, “Stochastic local operations with classical communication of absolutely maximally entangled states,” Phys. Rev. A 102, 022413 (2020).
  65. G. Kempf and L. Ness, “Algebraic geometry,”  (Springer, Berlin, Germany, 1979) Chap. The length of vectors in representation spaces, pp. 233–243.
  66. G. Gour and N. R. Wallach, “Necessary and sufficient conditions for local manipulation of multipartite pure quantum states,” New J. Phys. 13, 073013 (2011).
  67. M. Hayashi, D. Markham, M. Murao, M. Owari,  and S. Virmani, “Entanglement of multiparty-stabilizer, symmetric, and antisymmetric states,” Phys. Rev. A 77, 012104 (2008).
  68. D. Goyeneche, Z. Raissi, S. Di Martino,  and K. Życzkowsk, “Entanglement and quantum combinatorial designs,” Phys. Rev. A 97, 062326 (2018).
  69. W. Helwig, “Absolutely maximally entangled qudit graph states,” arXiv:1306.2879  (2013).
  70. E. M. Rains, “Nonbinary quantum codes,” IEEE Trans. Inf. Theory 45, 1827 (1999).
  71. M. Demianowicz and R. Augusiak, “From unextendible product bases to genuinely entangled subspaces,” Phys. Rev. A 98, 012313 (2018).
  72. K. R. Parthasarathy, “On the maximal dimension of a completely entangled subspace for finite level quantum systems,” Proc. Math. Sci. 114, 365–374 (2004).
  73. G. K. Pedersen, “Analysis now,”  (Springer New York, 1989) Chap. Hilbert Spaces, pp. 79–125.
  74. B. T. Polyak, “Some methods for speeding up the convergence of iteration methods,” USSR Comput. Math. and Math. Phys. 4, 1–17 (1964).
  75. T. Tilma and E. C. G. Sudarshan, “Generalized Euler angle parametrization for SU⁢(n)SU𝑛\text{SU}(n)SU ( italic_n ),” J. Math. Phys. 35, 10467 (2002).
  76. C. Jarlskog, “A recursive parametrization of unitary matrices,” J. Math. Phys. 46, 103508 (2005).
  77. C. Spengler, M. Huber,  and B. C. Hiesmayr, “A composite parameterization of unitary groups, density matrices and subspaces,” J. Phys. A 43, 385306 (2010).
  78. M. Hein, J. Eisert,  and H. J. Briegel, “Multiparty entanglement in graph states,” Phys. Rev. A 69, 062311 (2004).
  79. F. Verstraete and I. Cirac, “Matrix product states represent ground states faithfully,” Phys. Rev. B 73, 094423 (2006).
  80. H. Qassim, H. Pashayan,  and D. Gosset, “Improved upper bounds on the stabilizer rank of magic states,” Quantum 5, 606 (2021).
  81. T. Kraft, C. Ritz, N. Brunner, M. Huber,  and O. Gühne, “Characterizing Genuine Multilevel Entanglement,” Phys. Rev. Lett. 120, 060502 (2018).
  82. S. Bravyi, G. Smith,  and J. A. Smolin, “Trading Classical and Quantum Computational Resources,” Phys. Rev. X 6, 021043 (2016).
  83. T. Kraft, S. Designolle, C. Ritz, N. Brunner, O. Gühne,  and M. Huber, “Quantum entanglement in the triangle network,” Phys. Rev. A 103, L060401 (2021).
  84. T. G. Kolda and B. W. Bader, “Tensor Decompositions and Applications,” SIAM Review 51, 455–500 (2009).
  85. N. Yu, E. Chitambar, C. Guo,  and R. Duan, “Tensor rank of the tripartite state |W⊗2⟩ketsuperscriptWtensor-productabsent2|\text{W}^{\otimes 2}\rangle| W start_POSTSUPERSCRIPT ⊗ 2 end_POSTSUPERSCRIPT ⟩,” Phys. Rev. A 81, 014301 (2010).
  86. M. Erhard, M. Krenn,  and A. Zeilinger, “Advances in high-dimensional quantum entanglement,” Nat. Rev. Phys. 2, 365–381 (2020).
  87. L. Qi, G. Zhang,  and G. Ni, “How entangled can a multi-party system possibly be?” Phys. Lett. A 382, 1465–1471 (2018b).
  88. D. Gross, “Hudson’s theorem for finite-dimensional quantum systems,” J. Math. Phys. 47, 122107 (2006).
  89. J. Ja’Ja’, “Optimal evaluation of pairs of bilinear forms,” SIAM J. Comput. 8 (1979), 10.1137/0208037.
  90. A. Nico-Katz and S. Bose, “Entanglement-complexity geometric measure,” Phys. Rev. Research 5, 013041 (2023).
  91. D. Perez-Garcia, F. Verstraete, M. M. Wolf,  and J. I. Cirac, “Matrix product state representations,” Quantum Inf. Comput. 7, 401–430 (2007).
  92. G. Vidal, ‘‘Efficient classical simulation of slightly entangled quantum computations,” Phys. Rev. Lett. 91, 147902 (2003).
  93. S. Friedland and T. Kemp, “Most boson quantum states are almost maximally entangled,” Proc. Am. Math. Soc. 146, 5035–5049 (2018).
  94. K. Zyczkowski, M. Kus,  and H.-J. Sommers W. Słomczyński, “Random unistochastic matrices,” J. Phys. A 36, 3425 (2003).
  95. J. Eisert and H.-J. Briegel, “The Schmidt Measure as a Tool for Quantifying Multi-Particle Entanglement,” Phys. Rev. A 64, 022306 (2001).
  96. A. Uhlmann, “Fidelity and concurrence of conjugated states,” Phys. Rev. A 62, 032307 (2000).
  97. H. J. Kimble, “The quantum internet,” Nature 453, 1023 (2008).
  98. N. Sangouard, C. Simon, H. de Riedmatten,  and N. Gisin, “Quantum repeaters based on atomic ensembles and linear optics,” Rev. Mod. Phys. 83, 33 (2011).
  99. W. Hoeffding, “A Class of Statistics with Asymptotically Normal Distribution,” in Breakthroughs in Statistics, Springer Series in Statistics, edited by S. Kotz and N. L. Johnson (Springer, New York, 1992) pp. 308–334.
  100. K. Fitter, C. Lancien,  and I. Nechita, “Estimating the entanglement of random multipartite quantum states,” arXiv:2209.11754  (2022).
  101. D. Gottesman, “Theory of fault-tolerant quantum computation,” Phys. Rev. A 57, 127 (1998).

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
Youtube Logo Streamline Icon: https://streamlinehq.com

YouTube