Resource theories of multi-time processes: A window into quantum non-Markovianity

Graeme D. Berk; Andrew J. P. Garner; Benjamin Yadin; Kavan Modi; Felix A. Pollock

doi:10.22331/q-2021-04-20-435

Resource theories of multi-time processes: A window into quantum non-Markovianity

Graeme D. Berk¹, Andrew J. P. Garner^2,3, Benjamin Yadin⁴, Kavan Modi¹, and Felix A. Pollock¹

¹School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia
²Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria
³School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, 637371, Singapore
⁴School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom.

Published:	2021-04-20, volume 5, page 435
Eprint:	arXiv:1907.07003v4
Doi:	https://doi.org/10.22331/q-2021-04-20-435
Citation:	Quantum 5, 435 (2021).

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Abstract

We investigate the conditions under which an uncontrollable background processes may be harnessed by an agent to perform a task that would otherwise be impossible within their operational framework. This situation can be understood from the perspective of resource theory: rather than harnessing 'useful' quantum states to perform tasks, we propose a resource theory of quantum processes across multiple points in time. Uncontrollable background processes fulfil the role of resources, and a new set of objects called $\textit{superprocesses}$, corresponding to operationally implementable control of the system undergoing the process, constitute the transformations between them. After formally introducing a framework for deriving resource theories of multi-time processes, we present a hierarchy of examples induced by restricting quantum or classical communication within the superprocess — corresponding to a client-server scenario. The resulting nine resource theories have different notions of quantum or classical memory as the determinant of their utility. Furthermore, one of these theories has a strict correspondence between non-useful processes and those that are Markovian and, therefore, could be said to be a true 'quantum resource theory of non-Markovianity'.

Featured image: A restricted agent seeks to harness an uncontrolled background process (weather). They
achieve this via a transformation of that process (converting energy into electricity) such that the
resultant object can be utilised to perform a useful task (switching on a light-bulb). The utility of a background process is dependent on the tools an agent has to harness it.

Popular summary

In the olden times, sailors navigated the seas at the mercy of an uncontrolled background process, namely, the weather. Winds were their primary resource, and they developed sophisticated ways to harness winds to their advantage. In the modern era, another class of energy resources has risen to prominence, which unlike the wind, are directly controllable; these include fossil fuels, nuclear energy, and electric batteries.

Nascent quantum technologies are more akin to the sailboats of a bygone era as they are inevitably influenced by the environment they exist within. Our paper asks to what degree one can harness useful properties of this environment to their advantage. What is the quantum analogue of sailing? In particular, can we make use of patterns in the environment, i.e. the temporal correlations therein known as non-Markovianity?

Using a mathematical toolbox known as resource theories, we present a general framework for answering these types of questions, uncovering families of monotones — functions that measure the usefulness of resources. Additionally, we analysed how these rules play out for a hierarchy of nine example scenarios, varying the level of capability to harness an uncontrolled process. In one of these example scenarios we show that the uncontrolled processes that are useful correspond precisely to the aforementioned non-Markovianity property.

► BibTeX data

@article{Berk2021resourcetheoriesof,
  doi = {10.22331/q-2021-04-20-435},
  url = {https://doi.org/10.22331/q-2021-04-20-435},
  title = {Resource theories of multi-time processes: {A} window into quantum non-{M}arkovianity},
  author = {Berk, Graeme D. and Garner, Andrew J. P. and Yadin, Benjamin and Modi, Kavan and Pollock, Felix A.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {435},
  month = apr,
  year = {2021}
}

► References

[1] S. J. Devitt, W. J. Munro, and K. Nemoto, ``Quantum error correction for beginners,'' Rep. Prog. Phys 76, 076001 (2013).
https://doi.org/10.1088/0034-4885/76/7/076001

[2] L. Viola, E. Knill, and S. Lloyd, ``Dynamical decoupling of open quantum systems,'' Phys. Rev. Lett. 82, 2417 (1999).
https://doi.org/10.1103/PhysRevLett.82.2417

[3] F. G. S. L. Brandão, 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

[4] J. Goold, M. Huber, A. Riera, L. del Rio, and P. Skrzypczyk, ``The role of quantum information in thermodynamics—a topical review,'' J. Phys. A 49, 143001 (2016).
https://doi.org/10.1088/1751-8113/49/14/143001

[5] C. Sparaciari, J. Oppenheim, and T. Fritz, ``Resource theory for work and heat,'' Phys. Rev. A 96, 052112 (2017).
https://doi.org/10.1103/PhysRevA.96.052112

[6] P. Faist and R. Renner, ``Fundamental work cost of quantum processes,'' Phys. Rev. X 8, 021011 (2018).
https://doi.org/10.1103/PhysRevX.8.021011

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

[8] A. Streltsov, H. Kampermann, S. Wölk, M. Gessner, and D. Bruß, ``Maximal coherence and the resource theory of purity,'' New J. Phys 20, 053058 (2018).
https://doi.org/10.1088/1367-2630/aac484

[9] G. Gour, M. P. Muller, V. Narasimhachar, R. W. Spekkens, and N. Y. Halpern, ``The resource theory of informational nonequilibrium in thermodynamics,'' Phys. Rep. 583, 1 (2015).
https://doi.org/10.1016/j.physrep.2015.04.003

[10] 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

[11] F. G. S. L. Brandão and M. B. Plenio, ``Entanglement theory and the second law of thermodynamics,'' Nat. Phys. 4, 873 EP (2008).
https://doi.org/10.1038/nphys1100

[12] F. G. S. L. Brandão and G. Gour, ``Reversible framework for quantum resource theories,'' Phys. Rev. Lett. 115, 070503 (2015).
https://doi.org/10.1103/PhysRevLett.115.070503

[13] C. Sparaciari, L. del Rio, C. M. Scandolo, P. Faist, and J. Oppenheim, ``The first law of general quantum resource theories,'' (2018).
https://doi.org/10.22331/q-2020-04-30-259

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

[15] V. Veitch, S. A. H. Mousavian, D. Gottesman, and J. Emerson, ``The resource theory of stabilizer quantum computation,'' New J. Phys 16, 013009 (2014).
https://doi.org/10.1088/1367-2630/16/1/013009

[16] J. Morris, F. A. Pollock, and K. Modi, ``Non-Markovian memory in IBMQX4,'' arXiv:1902.07980 (2019).
arXiv:1902.07980

[17] A. W. Chin, S. F. Huelga, and M. B. Plenio, ``Quantum metrology in non-Markovian environments,'' Phys. Rev. Lett. 109, 233601 (2012).
https://doi.org/10.1103/PhysRevLett.109.233601

[18] D. M. Reich, N. Katz, and C. P. Koch, ``Exploiting non-Markovianity for quantum control,'' Sci. Rep. 5, 12430 (2015).
https://doi.org/10.1038/srep12430

[19] G. Thomas, N. Siddharth, S. Banerjee, and S. Ghosh, ``Thermodynamics of non-Markovian reservoirs and heat engines,'' Phys. Rev. E 97, 062108 (2018).
https://doi.org/10.1103/PhysRevE.97.062108

[20] S. Bhattacharya, B. Bhattacharya, and A. S. Majumdar, ``Thermodynamic utility of non-Markovianity from the perspective of resource interconversion,'' (2019).
https://doi.org/10.1088/1751-8121/aba0ee

[21] N. Erez, G. Gordon, M. Nest, and G. Kurizki, ``Thermodynamic control by frequent quantum measurements,'' Nature 452, 724 (2008).
https://doi.org/10.1038/nature06873

[22] B. Bylicka, D. Chruściński, and S. Maniscalco, ``Non-Markovianity as a resource for quantum technologies,'' arXiv:1301.2585 (2013).
arXiv:1301.2585

[23] N. Mirkin, P. Poggi, and D. Wisniacki, ``Information backflow as a resource for entanglement,'' (2019).
https://doi.org/10.1103/PhysRevA.99.062327

[24] B. Bylicka, D. Chruściński, and S. Maniscalco, ``Non-Markovianity and reservoir memory of quantum channels: a quantum information theory perspective,'' Sci. Rep. 4, 5720 (2014).
https://doi.org/10.1038/srep05720

[25] E.-M. Laine, H.-P. Breuer, and J. Piilo, ``Nonlocal memory effects allow perfect teleportation with mixed states,'' Sci. Rep. 4, 4620 (2014).
https://doi.org/10.1038/srep04620

[26] D. Rosset, F. Buscemi, and Y.-C. Liang, ``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

[27] E. Wakakuwa, ``Operational resource theory of non-Markovianity,'' arXiv:1709.07248 (2017).
arXiv:1709.07248

[28] S. Milz, F. A. Pollock, and K. Modi, ``An introduction to operational quantum dynamics,'' Open Syst. Inf. Dyn. 24, 1740016 (2017a).
https://doi.org/10.1142/S1230161217400169

[29] F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, ``Operational Markov condition for quantum processes,'' Phys. Rev. Lett. 120, 040405 (2018a).
https://doi.org/10.1103/PhysRevLett.120.040405

[30] F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, ``Non-Markovian quantum processes: complete framework and efficient characterization,'' Phys. Rev. A 97, 012127 (2018b).
https://doi.org/10.1103/PhysRevA.97.012127

[31] S. Milz, M. S. Kim, F. A. Pollock, and K. Modi, ``CP divisibility does not mean Markovianity,'' (2019).
https://doi.org/10.1103/PhysRevLett.123.040401

[32] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Theoretical framework for quantum networks,'' Phys. Rev. A 80, 022339 (2009).
https://doi.org/10.1103/PhysRevA.80.022339

[33] A. Bisio and P. Perinotti, ``Theoretical framework for higher-order quantum theory,'' Proc. R. Soc. A 475, 20180706 (2019).
https://doi.org/10.1098/rspa.2018.0706

[34] F. Sakuldee, S. Milz, F. A. Pollock, and K. Modi, ``Non-Markovian quantum control as coherent stochastic trajectories,'' J. Phys. A 51, 414014 (2018).
https://doi.org/10.1088/1751-8121/aabb1e

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

[36] S. Milz, F. Sakuldee, F. A. Pollock, and K. Modi, ``Kolmogorov extension theorem for (quantum) causal modelling and general probabilistic theories,'' (2017b).
https://doi.org/10.22331/q-2020-04-20-255

[37] F. Costa and S. Shrapnel, ``Quantum causal modelling,'' New J. Phys. 18, 063032 (2016).
https://doi.org/10.1088/1367-2630/18/6/063032

[38] K. Modi, ``Operational approach to open dynamics and quantifying initial correlations,'' Sci. Rep. 2, 581 (2012).
https://doi.org/10.1038/srep00581

[39] M. Ringbauer, C. J. Wood, K. Modi, A. Gilchrist, A. G. White, and A. Fedrizzi, ``Characterizing quantum dynamics with initial system-environment correlations,'' Phys. Rev. Lett. 114, 090402 (2015).
https://doi.org/10.1103/PhysRevLett.114.090402

[40] F. A. Pollock and K. Modi, ``Tomographically reconstructed master equations for any open quantum dynamics,'' Quantum 2, 76 (2018).
https://doi.org/10.22331/q-2018-07-11-76

[41] S. Shrapnel, F. Costa, and G. Milburn, ``Updating the Born rule,'' New J. Phys. 20, 053010 (2018).
https://doi.org/10.1088/1367-2630/aabe12

[42] S. Milz, F. A. Pollock, and K. Modi, ``Reconstructing non-Markovian quantum dynamics with limited control,'' Phys. Rev. A 98, 012108 (2018a).
https://doi.org/10.1103/PhysRevA.98.012108

[43] S. Shrapnel and F. Costa, ``Causation does not explain contextuality,'' Quantum 2, 63 (2018).
https://doi.org/10.22331/q-2018-05-18-63

[44] F. Costa, M. Ringbauer, M. E. Goggin, A. G. White, and A. Fedrizzi, ``Unifying framework for spatial and temporal quantum correlations,'' Phys. Rev. A 98, 012328 (2018).
https://doi.org/10.1103/PhysRevA.98.012328

[45] P. Taranto, S. Milz, F. A. Pollock, and K. Modi, ``Structure of quantum stochastic processes with finite Markov order,'' Phys. Rev. A 99, 042108 (2019a).
https://doi.org/10.1103/PhysRevA.99.042108

[46] P. Taranto, F. A. Pollock, S. Milz, M. Tomamichel, and K. Modi, ``Quantum Markov order,'' Phys. Rev. Lett. 122, 140401 (2019b).
https://doi.org/10.1103/PhysRevLett.122.140401

[47] I. A. Luchnikov, S. V. Vintskevich, and S. N. Filippov, ``Dimension truncation for open quantum systems in terms of tensor networks,'' (2018).
arXiv:1801.07418

[48] I. A. Luchnikov, S. V. Vintskevich, D. A. Grigoriev, and S. N. Filippov, ``Machine learning non-Markovian quantum dynamics,'' (2019a).
https://doi.org/10.1103/PhysRevLett.124.140502

[49] I. A. Luchnikov, S. V. Vintskevich, H. Ouerdane, and S. N. Filippov, ``Simulation complexity of open quantum dynamics: connection with tensor networks,'' Phys. Rev. Lett. 122, 160401 (2019b).
https://doi.org/10.1103/PhysRevLett.122.160401

[50] P. Strasberg, ``An operational approach to quantum stochastic thermodynamics,'' (2018).
https://doi.org/10.1103/PhysRevE.100.022127

[51] P. Figueroa-Romero, K. Modi, and F. A. Pollock, ``Equilibration on average of temporally non-local observables in quantum systems,'' (2019).
https://doi.org/10.1103/PhysRevE.102.032144

[52] P. Figueroa-Romero, K. Modi, and F. A. Pollock, ``Almost Markovian processes from closed dynamics,'' Quantum 3, 136 (2019).
https://doi.org/10.22331/q-2019-04-30-136

[53] P. Strasberg and A. Winter, ``Stochastic thermodynamics with arbitrary interventions,'' (2019).
https://doi.org/10.1103/PhysRevE.100.022135

[54] P. Strasberg, ``Repeated interactions and quantum stochastic thermodynamics at strong coupling,'' (2019).
https://doi.org/10.1103/PhysRevLett.123.180604

[55] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Transforming quantum operations: quantum supermaps,'' Europhys. Lett. 83, 30004 (2008b).
https://doi.org/10.1209/0295-5075/83/30004

[56] D. Kretschmann and R. F. Werner, ``Quantum channels with memory,'' Phys. Rev. A 72, 062323 (2005).
https://doi.org/10.1103/PhysRevA.72.062323

[57] F. Caruso, V. Giovannetti, C. Lupo, and S. Mancini, ``Quantum channels and memory effects,'' Rev. Mod. Phys. 86, 1203 (2014).
https://doi.org/10.1103/RevModPhys.86.1203

[58] C. Portmann, C. Matt, U. Maurer, R. Renner, and B. Tackmann, ``Causal boxes: quantum information-processing systems closed under composition,'' IEEE Trans. Inf. Theory 63, 3277 (2017).
https://doi.org/10.1109/TIT.2017.2676805

[59] L. Hardy, ``Operational general relativity: possibilistic, probabilistic, and quantum,'' arXiv:1608.06940 (2016).
arXiv:1608.06940

[60] L. Hardy, ``The operator tensor formulation of quantum theory,'' Phil. Trans. R. Soc. A 370, 3385 (2012).
https://doi.org/10.1098/rsta.2011.0326

[61] J. Cotler, C.-M. Jian, X.-L. Qi, and F. Wilczek, ``Superdensity operators for spacetime quantum mechanics,'' J. High Energy Phys 2018, 93 (2018).
https://doi.org/10.1007/JHEP09(2018)093

[62] O. Oreshkov, F. Costa, and Č. Brukner, ``Quantum correlations with no causal order,'' Nat. Commun. 3, 1092 (2012).
https://doi.org/10.1038/ncomms2076

[63] O. Oreshkov and C. Giarmatzi, ``Causal and causally separable processes,'' New J. Phys. 18, 093020 (2016).
https://doi.org/10.1088/1367-2630/18/9/093020

[64] S. Milz, F. A. Pollock, T. P. Le, G. Chiribella, and K. Modi, ``Entanglement, non-markovianity, and causal non-separability,'' New J. Phys. 20, 033033 (2018b).
https://doi.org/10.1088/1367-2630/aaafee

[65] N. Barnett and J. P. Crutchfield, ``Computational mechanics of input-output processes: structured transformations and the $\epsilon$-transducer,'' J. Stat. Phys. 161, 404 (2015).
https://doi.org/10.1007/s10955-015-1327-5

[66] J. Thompson, A. J. P. Garner, V. Vedral, and M. Gu, ``Using quantum theory to simplify input–output processes,'' npj Quantum Inf. 3, 6 (2017).
https://doi.org/10.1038/s41534-016-0001-3

[67] G. Gutoski, Quantum Strategies and Local Operations, Ph.D. thesis, - (2010).
arXiv:1003.0038

[68] G. Gutoski and J. Watrous, ``Toward a general theory of quantum games,'' in Proceedings of the Thirty-ninth Annual ACM Symposium on Theory of Computing, STOC '07 (ACM, New York, NY, USA, 2007) pp. 565–574.
https://doi.org/10.1145/1250790.1250873

[69] G. Gutoski, ``On a measure of distance for quantum strategies,'' J. Math. Phys 53, 032202 (2012).
https://doi.org/10.1063/1.3693621

[70] G. Gutoski, A. Rosmanis, and J. Sikora, ``Fidelity of quantum strategies with applications to cryptography,'' arXiv:1704.04033 (2017).
arXiv:1704.04033

[71] G. Lindblad, ``Non-markovian quantum stochastic processes and their entropy,'' Comm. Math. Phys. 65, 281 (1979).
https://projecteuclid.org:443/euclid.cmp/1103904877

[72] L. Accardi, ``Quantum stochastic processes,'' in Statistical Physics and Dynamical Systems: Rigorous Results, edited by J. Fritz, A. Jaffe, and D. Szász (Birkhäuser Boston, Boston, MA, 1985) pp. 285–302.
https://doi.org/10.1007/978-1-4899-6653-7_16

[73] L. Li, M. J. Hall, and H. M. Wiseman, ``Concepts of quantum non-markovianity: A hierarchy,'' Physics Reports 759, 1 (2018).
https://doi.org/10.1016/j.physrep.2018.07.001

[74] Z.-W. Liu and A. Winter, ``Resource theories of quantum channels and the universal role of resource erasure,'' arXiv:1904.04201 (2019).
arXiv:1904.04201

[75] Y. Liu and X. Yuan, ``Operational resource theory of quantum channels,'' (2019).
https://doi.org/10.1103/PhysRevResearch.2.012035

[76] 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

[77] G. Gour, ``Comparison of quantum channels by superchannels,'' IEEE Trans. Inf. Theory 65, 1 (2019).
https://doi.org/10.1109/TIT.2019.2907989

[78] G. Gour and M. M. Wilde, ``Entropy of a quantum channel,'' arXiv:1808.06980 (2018).
arXiv:1808.06980

[79] G. Gour and C. M. Scandolo, ``The Entanglement of a bipartite channel,'' arXiv:1907.02552 (2019).
arXiv:1907.02552

[80] S. Bäuml, S. Das, X. Wang, and M. M. Wilde, ``Resource theory of entanglement for bipartite quantum channels,'' arXiv:1907.04181 (2019).
arXiv:1907.04181

[81] X. Wang and M. M. Wilde, ``Resource theory of asymmetric distinguishability for quantum channels,'' (2019).
https://doi.org/10.1103/PhysRevResearch.1.033169

[82] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Probabilistic theories with purification,'' Phys. Rev. A 81, 062348 (2010).
https://doi.org/10.1103/PhysRevA.81.062348

[83] M. B. Ruskai, ``Beyond strong subadditivity? Improved bounds on the C=contraction of generalized relative entropy,'' Rev. Math. Phys. 6, 1147 (1994).
https://doi.org/10.1142/S0129055X94000407

[84] 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

[85] P. Skrzypczyk and N. Linden, ``Robustness of measurement, discrimination games, and accessible information,'' Phys. Rev. Lett. 122, 140403 (2019).
https://doi.org/10.1103/PhysRevLett.122.140403

[86] N. Datta, ``Max-relative entropy of entanglement, alias log robustness,'' International Journal of Quantum Information 07, 475 (2009).
https://doi.org/10.1142/S0219749909005298

[87] F. Leditzky, E. Kaur, N. Datta, and M. M. Wilde, ``Approaches for approximate additivity of the Holevo information of quantum channels,'' Phys. Rev. A 97, 012332 (2018).
https://doi.org/10.1103/PhysRevA.97.012332

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

[89] M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information: 10th anniversary edition (Cambridge University Press, 2010).
https://doi.org/10.1017/CBO9780511976667

[90] M. D'Angelo, A. Valencia, M. H. Rubin, and Y. Shih, ``Resolution of quantum and classical ghost imaging,'' Phys. Rev. A 72, 013810 (2005).
https://doi.org/10.1103/PhysRevA.72.013810

[91] J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O'Brien, M. A. Nielsen, and A. G. White, ``Ancilla-assisted quantum process tomography,'' Phys. Rev. Lett. 90, 193601 (2003).
https://doi.org/10.1103/PhysRevLett.90.193601

[92] P. Strasberg and M. García Díaz, ``Classical quantum stochastic processes,'' (2019).
https://doi.org/10.1103/PhysRevA.100.022120

[93] C. Giarmatzi and F. Costa, ``Witnessing quantum memory in non-Markovian processes,'' arXiv:1811.03722 (2018).
arXiv:1811.03722

[94] E. Chitambar, D. Leung, L. Mančinska, M. Ozols, and A. Winter, ``Everything you always wanted to know about LOCC (but were afraid to ask),'' Communications in Mathematical Physics 328, 303 (2014).
https://doi.org/10.1007/s00220-014-1953-9

[95] 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

[96] D. Gottesman, ``Theory of quantum secret sharing,'' Phys. Rev. A 61, 042311 (2000).
https://doi.org/10.1103/PhysRevA.61.042311

[97] R. Uola, T. Kraft, and A. A. Abbott, ``Quantification of quantum dynamics with input-output games,'' (2019).
https://doi.org/10.1103/PhysRevA.101.052306

[98] R. Takagi and B. Regula, ``General resource theories in quantum mechanics and beyond: operational characterization via discrimination tasks,'' (2019).
https://doi.org/10.1103/PhysRevX.9.031053

[99] S. Bäuml, S. Das, and M. M. Wilde, ``Fundamental limits on the capacities of bipartite quantum interactions,'' Phys. Rev. Lett. 121, 250504 (2018).
https://doi.org/10.1103/PhysRevLett.121.250504

[100] S. Das, ``Bipartite Quantum Interactions: Entangling and Information Processing Abilities,'' arXiv:1901.05895 (2019).
arXiv:1901.05895

[101] S. Das, S. Bäuml, and M. M. Wilde, ``Entanglement and secret-key-agreement capacities of bipartite quantum interactions and read-only memory devices,'' (2017).
https://doi.org/10.1103/PhysRevA.101.012344

[102] S. Bhattacharya, B. Bhattacharya, and A. S. Majumdar, ``Convex resource theory of non-Markovianity,'' arXiv:1803.06881 (2018).
arXiv:1803.06881

[103] N. Anand and T. A. Brun, ``Quantifying non-Markovianity: a quantum resource-theoretic approach,'' arXiv:1903.03880 (2019).
arXiv:1903.03880

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

[105] C.-C. Kuo, S.-H. Chen, W.-T. Lee, H.-M. Chen, H. Lu, and C.-M. Li, ``Quantum process capability,'' (2018).
https://doi.org/10.1038/s41598-019-56751-x

[106] K.-H. Wang, S.-H. Chen, Y.-C. Lin, and C.-M. Li, ``Non-Markovianity of photon dynamics in a birefringent crystal,'' Phys. Rev. A 98, 043850 (2018).
https://doi.org/10.1103/PhysRevA.98.043850

[107] E. Wakakuwa, ``Communication cost for non-Markovianity of tripartite quantum states: a resource theoretic approach,'' (2019).
https://doi.org/10.1109/TIT.2020.3028837

[108] T. Guff, N. A. McMahon, Y. R. Sanders, and A. Gilchrist, ``A resource theory of quantum measurements,'' arXiv:1902.08490 (2019).
arXiv:1902.08490

[109] R. Chaves, G. Carvacho, I. Agresti, V. Di Giulio, L. Aolita, S. Giacomini, and F. Sciarrino, ``Quantum violation of an instrumental test,'' Nat. Phys 14, 291 (2018).
https://doi.org/10.1038/s41567-017-0008-5

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