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Computable entanglement cost under positive partial transpose operations (2405.09613v2)

Published 15 May 2024 in quant-ph, cond-mat.stat-mech, math-ph, and math.MP

Abstract: Quantum information theory is plagued by the problem of regularisations, which require the evaluation of formidable asymptotic quantities. This makes it computationally intractable to gain a precise quantitative understanding of the ultimate efficiency of key operational tasks such as entanglement manipulation. Here we consider the problem of computing the asymptotic entanglement cost of preparing noisy quantum states under quantum operations with positive partial transpose (PPT). By means of an analytical example, a previously claimed solution to this problem is shown to be incorrect. Building on a previous characterisation of the PPT entanglement cost in terms of a regularised formula, we construct instead a hierarchy of semi-definite programs that bypasses the issue of regularisation altogether, and converges to the true asymptotic value of the entanglement cost. Our main result establishes that this convergence happens exponentially fast, thus yielding an efficient algorithm that approximates the cost up to an additive error $\varepsilon$ in time $\mathrm{poly}(D,\,\log(1/\varepsilon))$, where $D$ is the underlying Hilbert space dimension. To our knowledge, this is the first time that an asymptotic entanglement measure is shown to be efficiently computable despite no closed-form formula being available.

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References (41)
  1. A. S. Holevo, The capacity of the quantum channel with general signal states, IEEE Trans. Inf. Theory 44, 269 (1998).
  2. B. Schumacher and M. D. Westmoreland, Sending classical information via noisy quantum channels, Phys. Rev. A 56, 131 (1997).
  3. S. Lloyd, Capacity of the noisy quantum channel, Phys. Rev. A 55, 1613 (1997).
  4. P. Shor, The quantum channel capacity and coherent information, Lecture notes, MSRI Workshop on Quantum Computation (2002).
  5. I. Devetak, The private classical capacity and quantum capacity of a quantum channel, IEEE Trans. Inf. Theory 51, 44 (2005).
  6. I. Devetak and A. Winter, Distillation of secret key and entanglement from quantum states, Proc. Royal Soc. A 461, 207 (2005).
  7. L. Lami and B. Regula, Distillable entanglement under dually non-entangling operations, Preprint arXiv:2307.11008 (2023a).
  8. K. G. H. Vollbrecht and R. F. Werner, Entanglement measures under symmetry, Phys. Rev. A 64, 062307 (2001).
  9. P. W. Shor, Equivalence of additivity questions in quantum information theory, Commun. Math. Phys. 246, 473 (2004).
  10. P. Hayden and A. Winter, Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1, Commun. Math. Phys. 284, 263 (2008).
  11. M. B. Hastings, Superadditivity of communication capacity using entangled inputs, Nat. Phys. 5, 255 (2009).
  12. G. Smith and J. Yard, Quantum communication with zero-capacity channels, Science 321, 1812 (2008).
  13. W. K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Phys. Rev. Lett. 80, 2245 (1998).
  14. X. Wang and M. M. Wilde, Cost of quantum entanglement simplified, Phys. Rev. Lett. 125, 040502 (2020).
  15. X. Wang and M. M. Wilde, Exact entanglement cost of quantum states and channels under positive-partial-transpose-preserving operations, Phys. Rev. A 107, 012429 (2023).
  16. G. Gour and C. M. Scandolo, Dynamical entanglement, Phys. Rev. Lett. 125, 180505 (2020).
  17. G. Gour and C. M. Scandolo, Entanglement of a bipartite channel, Phys. Rev. A 103, 062422 (2021).
  18. S. Ishizaka, Binegativity and geometry of entangled states in two qubits, Phys. Rev. A 69, 020301 (2004).
  19. A. Peres, Separability criterion for density matrices, Phys. Rev. Lett. 77, 1413 (1996).
  20. G. Vidal and R. F. Werner, Computable measure of entanglement, Phys. Rev. A 65, 032314 (2002).
  21. M. B. Plenio, Logarithmic negativity: A full entanglement monotone that is not convex, Phys. Rev. Lett. 95, 090503 (2005).
  22. J. Watrous, The Theory of Quantum Information (Cambridge University Press, 2018).
  23. P. Skrzypczyk and D. Cavalcanti, Semidefinite Programming in Quantum Information Science (IOP Publishing, 2023).
  24. L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Rev. 38, 49 (1996).
  25. X. Wang and M. M. Wilde, Errata for “Cost of quantum entanglement simplified” and “Exact entanglement cost of quantum states and channels under PPT-preserving operations”, preprint https://doi.org/10.5281/zenodo.11061607 (2024).
  26. See the Supplemental Material.
  27. H. Fawzi and O. Fawzi, Defining quantum divergences via convex optimization, Quantum 5, 387 (2021).
  28. K. Fang and H. Fawzi, Geometric Rényi divergence and its applications in quantum channel capacities, Commun. Math. Phys. 384, 1615 (2021).
  29. E. M. Rains, Bound on distillable entanglement, Phys. Rev. A 60, 179 (1999).
  30. E. M. Rains, A semidefinite program for distillable entanglement, IEEE Trans. Inf. Theory 47, 2921 (2001).
  31. X. Wang and R. Duan, Irreversibility of asymptotic entanglement manipulation under quantum operations completely preserving positivity of partial transpose, Phys. Rev. Lett. 119, 180506 (2017).
  32. R. F. Werner, Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model, Phys. Rev. A 40, 4277 (1989).
  33. M. Horodecki, P. Horodecki,  and R. Horodecki, Mixed-state entanglement and distillation: Is there a “bound” entanglement in nature? Phys. Rev. Lett. 80, 5239 (1998).
  34. D. Bruß and A. Peres, Construction of quantum states with bound entanglement, Phys. Rev. A 61, 030301 (2000).
  35. M. Fekete, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Z. 17, 228 (1923).
  36. M. A. Nielsen, Conditions for a class of entanglement transformations, Phys. Rev. Lett. 83, 436 (1999).
  37. S. Khatri and M. M. Wilde, Principles of quantum communication theory: A modern approach, preprint arXiv:2011.04672 (2020).
  38. M. Horodecki, Entanglement measures, Quantum Inf. Comput. 1, 3 (2001).
  39. L. Lami and B. Regula, No second law of entanglement manipulation after all, Nat. Phys. 19, 184 (2023b).
  40. G. Vidal and R. Tarrach, Robustness of entanglement, Phys. Rev. A 59, 141 (1999).
  41. C. Shannon, The zero error capacity of a noisy channel, IRE Trans. Inf. Theory 2, 8 (1956).
Citations (1)
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Summary

  • The paper refutes prior claims by showing that the measure Eₖ is non-additive via a counterexample.
  • It introduces two SDP hierarchies that bound the entanglement cost from below and above, achieving exponential convergence.
  • The efficient algorithm developed paves the way for improved quantum protocols and further research in quantum resource management.

Overview of the Paper on Computable Entanglement Cost

The paper "Computable Entanglement Cost" by Lami, Mele, and Regula addresses the challenge of computing the entanglement cost of preparing quantum states under positive partial transpose (PPT) operations. This topic is situated within quantum information theory, where entanglement manipulation and quantification are central challenges, exacerbated by the regularisation problem—requiring asymptotic evaluations that hinder computational tractability.

Main Contributions

  1. Verification of Previous Claims: The authors refute a previous claim regarding the computability of the entanglement cost using a quantity called EκE_\kappa. They construct a counterexample demonstrating that EκE_\kappa is not additive, contradicting prior assertions that it could exactly characterise the entanglement cost without regularisation.
  2. Two Hierarchies of Semi-definite Programs (SDPs): The authors propose two sequences of SDPs—the χ\chi-hierarchy and the κ\kappa-hierarchy—that approximate the entanglement cost from below and above, respectively. This approach permits bridging the gap left by the previous non-additivity of EκE_\kappa.
  3. Efficient Algorithm for Computation: The core technical achievement is proving that the χ\chi-hierarchy converges exponentially fast to the exact entanglement cost. This finding leads to an algorithm that approximates the cost efficiently—the first known method for a meaningful operational entanglement measure.

Explored Concepts

  • Entanglement Cost Under PPT Operations: Understanding the amount of entanglement needed to create specific quantum states using PPT operations is crucial. PPT operations serve as a broader set than LOCC, allowing more general transformations while maintaining computational tractability.
  • Non-additivity of EκE_\kappa: The authors rigorously disprove the additivity of this proposed measure, using illustrative counterexamples, thereby highlighting a fundamental flaw in previous works related to EκE_\kappa.
  • Hierarchical Frameworks and Convergence: By presenting the χ\chi- and κ\kappa-hierarchies, the paper offers a new framework to estimate entanglement measures efficiently. The central technical observation of exponential convergence facilitates practical evaluation, overcoming the omnipresent curse of regularisation in quantum information theory.

Practical Implications and Future Directions

This work has broad implications for both theoretical understanding and practical computation within quantum information science. By addressing the computability of the entanglement cost, and thereby opening the door to precise entanglement management in quantum systems, the authors pave the way for improved protocols in quantum computing and communication.

Future research might explore whether these hierarchies can uncover similar efficiencies in other quantum measures, such as entanglement of formation or distillable entanglement. Furthermore, the findings encourage revisiting other regularisation-drained problems in quantum information theory, potentially extending the proposed framework beyond entanglement to other quantum resources.

Conclusion

The paper provides a decisive step forward in the computation of asymptotic quantum entanglement measures, moving past entrenched theoretical barriers. It balances the depth of theoretical exploration with the crafting of pragmatic algorithms, offering both a conceptual advance and a computational toolset for the quantum information community. The deeper understanding of non-additivity and the introduction of SDP hierarchies signify crucial developments that hold promise for further theoretical breakthroughs and practical implementations in quantum systems.

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