Resource Preservability

Chung-Yun Hsieh

doi:10.22331/q-2020-03-19-244

Resource Preservability

ICFO - Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Spain

Published:	2020-03-19, volume 4, page 244
Eprint:	arXiv:1910.02464v4
Doi:	https://doi.org/10.22331/q-2020-03-19-244
Citation:	Quantum 4, 244 (2020).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

Resource theory is a general, model-independent approach aiming to understand the qualitative notion of resource quantitatively. In a given resource theory, free operations are physical processes that do not create the resource and are considered zero-cost. This brings the following natural question: For a given free operation, what is its ability to preserve a resource? We axiomatically formulate this ability as the $\textit{resource preservability}$, which is constructed as a channel resource theory induced by a state resource theory. We provide two general classes of resource preservability monotones: One is based on state resource monotones, and another is based on channel distance measures. Specifically, the latter gives the robustness monotone, which has been recently found to have an operational interpretation. As examples, we show that athermality preservability of a Gibbs-preserving channel can be related to the smallest bath size needed to thermalize all its outputs, and it also bounds the capacity of a classical communication scenario under certain thermodynamic constraints. We further apply our theory to the study of entanglement preserving local thermalization (EPLT) and provide a new family of EPLT which admits arbitrarily small nonzero entanglement preservability and free entanglement preservation at the same time. Our results give the first systematic and general formulation of the resource preservation character of free operations.

► BibTeX data

@article{Hsieh2020resource,
  doi = {10.22331/q-2020-03-19-244},
  url = {https://doi.org/10.22331/q-2020-03-19-244},
  title = {Resource {P}reservability},
  author = {Hsieh, Chung-Yun},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {244},
  month = mar,
  year = {2020}
}

► References

[1] R, Horodecki, P, Horodecki, M, Horodecki, and K, Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
https://doi.org/10.1103/RevModPhys.81.865

[2] A. Peres, Separability criterion for density matrices, Phys. Rev. Lett. 77, 1413 (1996).
https://doi.org/10.1103/PhysRevLett.77.1413

[3] M. Horodecki, P. Horodecki, and R. Horodecki, Separability of mixed states: Necessary and sufficient conditions, Phys. Lett. A 223, 1 (1996).
https://doi.org/10.1016/S0375-9601(96)00706-2

[4] J. S. Bell, On the Einstein Podolsky Rosen paradox, Physics Physique Fizika 1, 195 (1964).
https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195

[5] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014).
https://doi.org/10.1103/RevModPhys.86.419

[6] H. M. Wiseman, S. J. Jones, and A. C. Doherty, Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox, Phys. Rev. Lett. 98, 140402 (2007).
https://doi.org/10.1103/PhysRevLett.98.140402

[7] S. J. Jones, H. M. Wiseman, and A. C. Doherty, Entanglement, Einstein-Podolsky-Rosen correlations, Bell nonlocality, and steering, Phys. Rev. A 76, 052116 (2007).
https://doi.org/10.1103/PhysRevA.76.052116

[8] D. Cavalcanti and P. Skrzypczyk, Quantum steering: A review with focus on semidefinite programming, Rep. Prog. Phys. 80, 024001 (2017).
https://doi.org/10.1088/1361-6633/80/2/024001

[9] R. Uola, A. C. S. Costa, H. C. Nguyen, and O. G$\ddot{\rm u}$hne, Quantum steering, Rev. Mod. Phys. 92, 015001 (2020).
https://doi.org/10.1103/RevModPhys.92.015001

[10] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett. 70, 1895 (1993).
https://doi.org/10.1103/PhysRevLett.70.1895

[11] M. Horodecki, P. Horodecki, and R. Horodecki, General teleportation channel, singlet fraction, and quasidistillation, Phys. Rev. A 60, 1888 (1999).
https://doi.org/10.1103/PhysRevA.60.1888

[12] D. Jennings and T. Rudolph, Entanglement and the thermodynamic arrow of time , Phys. Rev. E 81, 061130 (2010).
https://doi.org/10.1103/PhysRevE.81.061130

[13] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press, Cambridge, UK, 2000).

[14] V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Quantifying entanglement, Phys. Rev. Lett. 78, 2275 (1997).
https://doi.org/10.1103/PhysRevLett.78.2275

[15] A. Streltsov, G. Adesso, and M. B. Plenio, Colloquium: Quantum coherence as a resource, Rev. Mod. Phys. 89, 041003 (2017).
https://doi.org/10.1103/RevModPhys.89.041003

[16] T. Baumgratz, M. Cramer, and M. B. Plenio, Quantifying coherence, Phys. Rev. Lett. 113, 140401 (2014).
https://doi.org/10.1103/PhysRevLett.113.140401

[17] E. Wolfe, D. Schmid, A. B. Sainz, R. Kunjwal, and R. W. Spekkens, Quantifying Bell: The resource theory of nonclassicality of common-cause boxes, arXiv:1903.06311.
arXiv:1903.06311

[18] P. Skrzypczyk, M. Navascués, and D. Cavalcanti, Quantifying Einstein-Podolsky-Rosen steering, Phys. Rev. Lett. 112, 180404 (2014).
https://doi.org/10.1103/PhysRevLett.112.180404

[19] M. Piani and J. Watrous, Necessary and sufficient quantum information characterization of Einstein-Podolsky-Rosen steering, Phys. Rev. Lett. 114, 060404 (2015).
https://doi.org/10.1103/PhysRevLett.114.060404

[20] R. Gallego and L. Aolita, Resource theory of steering, Phys. Rev. X 5, 041008 (2015).
https://doi.org/10.1103/PhysRevX.5.041008

[21] G. Gour and R. W. Spekkens, The resource theory of quantum reference frames: Manipulations and monotones, New J. Phys. 10, 033023 (2008).
https://doi.org/10.1088/1367-2630/10/3/033023

[22] I. Marvian and R. W. Spekkens, How to quantify coherence: Distinguishing speakable and unspeakable notions, Phys. Rev. A 94, 052324 (2016).
https://doi.org/10.1103/PhysRevA.94.052324

[23] F. G. S. L. Brand$\tilde{\rm a}$o, M. Horodecki, J. Oppenheim, J. M. Renes, and R. W. Spekkens, Resource theory of quantum states out of thermal equilibrium, Phys. Rev. Lett. 111, 250404 (2013).
https://doi.org/10.1103/PhysRevLett.111.250404

[24] M. Horodecki and J. Oppenheim, Fundamental limitations for quantum and nanoscale thermodynamics, Nat. Commun. 4, 2059 (2013).
https://doi.org/10.1038/ncomms3059

[25] F. G. S. L. Brand$\tilde{\rm a}$o, M. Horodecki, N. Ng, J. Oppenheim, and S. Wehner, The second laws of quantum thermodynamics, Proc. Natl. Acad. Sci. U.S.A. 112, 3275 (2015).
https://doi.org/10.1073/pnas.1411728112

[26] M. Lostaglio, An introductory review of the resource theory approach to thermodynamics, Rep. Prog. Phys. 82, 114001 (2019).
https://doi.org/10.1088/1361-6633/ab46e5

[27] V. Narasimhachar, S. Assad, F. C. Binder, J. Thompson, B. Yadin, and M. Gu, Thermodynamic resources in continuous-variable quantum systems, arXiv:1909.07364.
arXiv:1909.07364

[28] A. Serafini, M. Lostaglio, S. Longden, U. Shackerley-Bennett, C.-Y. Hsieh, and G. Adesso, Gaussian thermal operations and the limits of algorithmic cooling, Phys. Rev. Lett. 124, 010602 (2020).
https://doi.org/10.1103/PhysRevLett.124.010602

[29] M. Horodecki and J. Oppenheim, (Quantumness in the context of) resource theories, Int. J. Mod. Phys. B 27, 1345019 (2013).
https://doi.org/10.1142/S0217979213450197

[30] F. G. S. L. Brand$\tilde{\rm a}$o and G. Gour, Reversible framework for quantum resource theories, Phys. Rev. Lett. 115, 070503 (2015).
https://doi.org/10.1103/PhysRevLett.115.070503

[31] L. del Rio, L. Kraemer, and R. Renner, Resource theories of knowledge, arXiv:1511.08818.
arXiv:1511.08818

[32] B. Coecke, T. Fritz, and R. W. Spekkens, A mathematical theory of resources, Inf. Comput. 250, 59 (2016).
https://doi.org/10.1016/j.ic.2016.02.008

[33] G. Gour, Quantum resource theories in the single-shot regime, Phys. Rev. A 95, 062314 (2017).
https://doi.org/10.1103/PhysRevA.95.062314

[34] Z.-W. Liu, X. Hu, and S. Lloyd, Resource destroying maps, Phys. Rev. Lett. 118, 060502 (2017).
https://doi.org/10.1103/PhysRevLett.118.060502

[35] A. Anshu, M.-H. Hsieh, and R. Jain, Quantifying resources in general resource theory with catalysts, Phys. Rev. Lett. 121, 190504 (2018).
https://doi.org/10.1103/PhysRevLett.121.190504

[36] E. Chitambar and G. Gour, Quantum resource theories, Rev. Mod. Phys. 91, 025001 (2019).
https://doi.org/10.1103/RevModPhys.91.025001

[37] L. Lami, B. Regula, X. Wang, R. Nichols, A. Winter, and G. Adesso, Gaussian quantum resource theories, Phys. Rev. A 98, 022335 (2018).
https://doi.org/10.1103/PhysRevA.98.022335

[38] B. Regula, Convex geometry of quantum resource quantification, J. Phys. A: Math. Theor. 51, 045303 (2018).
https://doi.org/10.1088/1751-8121/aa9100

[39] Z.-W. Liu, K. Bu, and R. Takagi, One-shot operational quantum resource theory, Phys. Rev. Lett. 123, 020401 (2019).
https://doi.org/10.1103/PhysRevLett.123.020401

[40] K. Fang and Z.-W. Liu, No-go theorems for quantum resource purification. arXiv:1909.02540.
arXiv:1909.02540

[41] R. Takagi and B. Regula, General resource theories in quantum mechanics and beyond: Operational characterization via discrimination tasks, Phys. Rev. X 9, 031053 (2019).
https://doi.org/10.1103/PhysRevX.9.031053

[42] R. Takagi, B. Regula, K. Bu, Z.-W. Liu, and G. Adesso, Operational advantage of quantum resources in subchannel discrimination, Phys. Rev. Lett. 122, 140402 (2019).
https://doi.org/10.1103/PhysRevLett.122.140402

[43] L. Li, K. Bu and Z.-W. Liu, Quantifying the resource content of quantum channels: An operational approach, Phys. Rev. A 101, 022335 (2020).
https://doi.org/10.1103/PhysRevA.101.022335

[44] J.-H. Hsieh, S.-H. Chen, and C.-M. Li, Quantifying quantum-mechanical processes, Scientific Reports 7, 13588 (2017).
https://doi.org/10.1038/s41598-017-13604-9

[45] C.-C. Kuo, S.-H. Chen, W.-T. Lee, H.-M. Chen, H. Lu, and C.-M. Li, Quantum process capability, Scientific Reports 9, 20316 (2019).
https://doi.org/10.1038/s41598-019-56751-x

[46] K. B. Dana, M. G. Díaz, M. Mejatty, and A. Winter, Resource theory of coherence: Beyond states, Phys. Rev. A 95, 062327 (2017).
https://doi.org/10.1103/PhysRevA.95.062327

[47] S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, Fundamental limits of repeaterless quantum communications, Nat. Commun. 8, 15043 (2017).
https://doi.org/10.1038/ncomms15043

[48] M. G. Díaz, K. Fang, X. Wang, M. Rosati, M. Skotiniotis, J. Calsamiglia, and A. Winter, Using and reusing coherence to realize quantum processes, Quantum 2, 100 (2018).
https://doi.org/10.22331/q-2018-10-19-100

[49] D. Rosset, F. Buscemi, and Y.-C. Liang, A resource theory of quantum memories and their faithful verification with minimal assumptions, Phys. Rev. X 8, 021033 (2018).
https://doi.org/10.1103/PhysRevX.8.021033

[50] M. M. Wilde, Entanglement cost and quantum channel simulation, Phys. Rev. A 98, 042338 (2018).
https://doi.org/10.1103/PhysRevA.98.042338

[51] Q. Zhuang, P. W. Shor, and J. H. Shapiro, Resource theory of non-Gaussian operations, Phys. Rev. A 97, 052317 (2018).
https://doi.org/10.1103/PhysRevA.97.052317

[52] S. B$\ddot{\rm a}$uml, S. Das, X. Wang, and M. M. Wilde, Resource theory of entanglement for bipartite quantum channels, arXiv:1907.04181.
arXiv:1907.04181

[53] J. R. Seddon and E. Campbell, Quantifying magic for multi-qubit operations, Proc. R. Soc. A 475, 20190251 (2019).
https://doi.org/10.1098/rspa.2019.0251

[54] Z.-W. Liu and A. Winter, Resource theories of quantum channels and the universal role of resource erasure, arXiv:1904.04201.
arXiv:1904.04201

[55] Y. Liu and X. Yuan, Operational resource theory of quantum channels, Phys. Rev. Research 2, 012035(R) (2020).
https://doi.org/10.1103/PhysRevResearch.2.012035

[56] G. Gour, Comparison of quantum channels by superchannels, IEEE Trans. Inf. Theory 65, 5880 (2019).
https://doi.org/10.1109/TIT.2019.2907989

[57] G. Gour and A. Winter, How to quantify a dynamical resource? Phys. Rev. Lett. 123, 150401 (2019).
https://doi.org/10.1103/PhysRevLett.123.150401

[58] G. Gour and C. M. Scandolo, The entanglement of a bipartite channel, arXiv:1907.02552.
arXiv:1907.02552

[59] R. Takagi, K. Wang, and M. Hayashi, Application of a resource theory of channels to communication scenarios, arXiv:1910.01125v1.
arXiv:1910.01125

[60] T. Theurer, D. Egloff, L. Zhang, and M. B. Plenio, Quantifying operations with an application to coherence, Phys. Rev. Lett. 122, 190405 (2019).
https://doi.org/10.1103/PhysRevLett.122.190405

[61] X. Wang and M. M. Wilde, Resource theory of asymmetric distinguishability for quantum channels, Phys. Rev. Research 1, 033169 (2019).
https://doi.org/10.1103/PhysRevResearch.1.033169

[62] G. D. Berk, A. J. P. Garner, B. Yadin, K. Modi, and F. A. Pollock, Resource theories of multi-time processes: A window into quantum non-Markovianity, arXiv:1907.07003.
arXiv:1907.07003

[63] C.-Y. Hsieh, M. Lostaglio, and A. Acín, Entanglement preserving local thermalization, Phys. Rev. Research 2, 013379 (2020).
https://doi.org/10.1103/PhysRevResearch.2.013379

[64] L. Morav${\check{\rm c}}$íková and M. Ziman, Entanglement-annihilating and entanglement-breaking channels, J Phys. A: Math. Theor. 43, 275306 (2010).
https://doi.org/10.1088/1751-8113/43/27/275306

[65] 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).
https://doi.org/10.1103/PhysRevLett.80.5239

[66] G. Chiribella, G. M. D'Ariano, and P. Perinotti, Transforming quantum operations: Quantum supermaps, EPL (Europhysics Letters) 83, 30004 (2008).
https://doi.org/10.1209/0295-5075/83/30004

[67] G. Chiribella, G. M. D'Ariano, and P. Perinotti, Quantum circuit architecture, Phys. Rev. Lett. 101, 060401 (2008).
https://doi.org/10.1103/PhysRevLett.101.060401

[68] C. Palazuelos, Superactivation of quantum nonlocality, Phys. Rev. Lett. 109, 190401 (2012).
https://doi.org/10.1103/PhysRevLett.109.190401

[69] C.-Y. Hsieh, Y.-C. Liang, and R.-K. Lee, Quantum steerability: Characterization, quantification, superactivation and unbounded amplification, Phys. Rev. A 94, 062120 (2016).
https://doi.org/10.1103/PhysRevA.94.062120

[70] M. T. Quintino, M. Huber, and N. Brunner, Superactivation of quantum steering, Phys. Rev. A 94, 062123 (2016).
https://doi.org/10.1103/PhysRevA.94.062123

[71] Ll. Masanes, Y.-C. Liang, and A. C. Doherty, All bipartite entangled states display some hidden nonlocality, Phys. Rev. Lett. 100, 090403 (2008).
https://doi.org/10.1103/PhysRevLett.100.090403

[72] Y.-C. Liang, Ll. Masanes, and D. Rosset, All entangled states display some hidden nonlocality, Phys. Rev. A 86, 052115 (2012).
https://doi.org/10.1103/PhysRevA.86.052115

[73] M. Horodecki, P. W. Shor, and M. B. Ruskai, Entanglement breaking channels, Rev. Math. Phys. 15, 629 (2003).
https://doi.org/10.1142/S0129055X03001709

[74] N. Datta, Min- and max-relative entropies and a new entanglement monotone, IEEE Trans. Inf. Theory 55, 2816 (2009).
https://doi.org/10.1109/TIT.2009.2018325

[75] G. Saxena, E. Chitambar,and G. Gour, Dynamical resource theory of quantum coherence, arXiv:1910.00708.
arXiv:1910.00708

[76] C. Sparaciari, M. Goihl, P. Boes, J. Eisert, and N. Ng, Bounding the resources for thermalizing many-body localized systems, arXiv:1912.04920.
arXiv:1912.04920

[77] D. Cavalcanti, A. Acin, N. Brunner, and T. Vertesi, All quantum states useful for teleportation are nonlocal resources, Phys. Rev. A 87, 042104 (2013).
https://doi.org/10.1103/PhysRevA.87.042104

[78] S. Albeverio, S.-M. Fei, and W.-L. Yang, Optimal teleportation based on bell measurements, Phys. Rev. A 66, 012301 (2002).
https://doi.org/10.1103/PhysRevA.66.012301

[79] M.-J. Zhao, Z.-G. Li, S.-M. Fei, and Z.-X. Wang, A note on fully entangled fraction, J. Phys. A: Math. Theor. 43, 275203 (2010).
https://doi.org/10.1088/1751-8113/43/27/275203

[80] C.-Y. Hsieh and R.-K. Lee, Work extraction and fully entangled fraction, Phys. Rev. A 96, 012107 (2017).
https://doi.org/10.1103/PhysRevA.96.012107

[81] Y.-C. Liang, Y.-H. Yeh, P. E. M. F. Mendonça, R. Y. Teh, M. D. Reid, and P. D. Drummond, Quantum fidelity measures for mixed states, Rep. Prog. Phys. 82, 076001 (2019).
https://doi.org/10.1088/1361-6633/ab1ca4

[82] M. Horodecki and P. Horodecki, Reduction criterion of separability and limits for a class of distillation protocols, Phys. Rev. A 59, 4206 (1999).
https://doi.org/10.1103/PhysRevA.59.4206

[83] C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, Purification of noisy entanglement and faithful teleportation via noisy channels, Phys. Rev. Lett. 76, 722 (1996).
https://doi.org/10.1103/PhysRevLett.76.722

[84] M. L. Almeida, S. Pironio, J. Barrett, G. Tóth, and A. Acín, Noise robustness of the nonlocality of entangled quantum states, Phys. Rev. Lett. 99, 040403 (2007).
https://doi.org/10.1103/PhysRevLett.99.040403

Cited by

[1] Jaehak Lee, Kyunghyun Baek, Jiyong Park, Jaewan Kim, and Hyunchul Nha, "Fundamental limits on concentrating and preserving tensorized quantum resources", Physical Review Research 4 4, 043070 (2022).

[2] Wei‐Hao Huang, Shih‐Hsuan Chen, Chun‐Hao Chang, Tzu‐Liang Hsu, Kuan‐Jou Wang, and Che‐Ming Li, "Quantum Correlation Generation Capability of Experimental Processes", Advanced Quantum Technologies 2300113 (2023).

[3] Huan-Yu Ku, Kuan-Yi Lee, Po-Rong Lai, Jhen-Dong Lin, and Yueh-Nan Chen, "Coherent activation of a steerability-breaking channel", Physical Review A 107 4, 042415 (2023).

[4] Ho-Joon Kim and Soojoon Lee, "Relation between quantum coherence and quantum entanglement in quantum measurements", Physical Review A 106 2, 022401 (2022).

[5] Bartosz Regula, "Tight constraints on probabilistic convertibility of quantum states", Quantum 6, 817 (2022).

[6] Huan-Yu Ku, Josef Kadlec, Antonín Černoch, Marco Túlio Quintino, Wenbin Zhou, Karel Lemr, Neill Lambert, Adam Miranowicz, Shin-Liang Chen, Franco Nori, and Yueh-Nan Chen, "Quantifying Quantumness of Channels Without Entanglement", PRX Quantum 3 2, 020338 (2022).

[7] Chung-Yun Hsieh, Matteo Lostaglio, and Antonio Acín, "Entanglement preserving local thermalization", Physical Review Research 2 1, 013379 (2020).

[8] Chung-Yun Hsieh, "Communication, Dynamical Resource Theory, and Thermodynamics", PRX Quantum 2 2, 020318 (2021).

[9] Shih-Hsuan Chen, Meng-Lok Ng, and Che-Ming Li, "Quantifying entanglement preservability of experimental processes", Physical Review A 104 3, 032403 (2021).

[10] Bartosz Regula, "Probabilistic Transformations of Quantum Resources", Physical Review Letters 128 11, 110505 (2022).

[11] Chung-Yun Hsieh, Matteo Lostaglio, and Antonio Acín, "Quantum channel marginal problem", Physical Review Research 4 1, 013249 (2022).

[12] Gaurav Saxena, Eric Chitambar, and Gilad Gour, "Dynamical resource theory of quantum coherence", Physical Review Research 2 2, 023298 (2020).

[13] Thomas Theurer, Saipriya Satyajit, and Martin B. Plenio, "Quantifying Dynamical Coherence with Dynamical Entanglement", Physical Review Letters 125 13, 130401 (2020).

The above citations are from Crossref's cited-by service (last updated successfully 2023-09-28 06:51:06) and SAO/NASA ADS (last updated successfully 2023-09-28 06:51:07). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.