Consequences of preserving reversibility in quantum superchannels

Wataru Yokojima; Marco Túlio Quintino; Akihito Soeda; Mio Murao

doi:10.22331/q-2021-04-26-441

Consequences of preserving reversibility in quantum superchannels

Wataru Yokojima, Marco Túlio Quintino, Akihito Soeda, and Mio Murao

Department of Physics, Graduate School of Science, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan

Published:	2021-04-26, volume 5, page 441
Eprint:	arXiv:2003.05682v5
Doi:	https://doi.org/10.22331/q-2021-04-26-441
Citation:	Quantum 5, 441 (2021).

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Abstract

Similarly to quantum states, quantum operations can also be transformed by means of quantum superchannels, also known as process matrices. Quantum superchannels with multiple slots are deterministic transformations which take independent quantum operations as inputs. While they are enforced to respect the laws of quantum mechanics, the use of input operations may lack a definite causal order, and characterizations of general superchannels in terms of quantum objects with a physical implementation have been missing. In this paper, we provide a mathematical characterization for pure superchannels with two slots (also known as bipartite pure processes), which are superchannels preserving the reversibility of quantum operations. We show that the reversibility preserving condition restricts all pure superchannels with two slots to be either a quantum circuit only consisting of unitary operations or a coherent superposition of two unitary quantum circuits where the two input operations are differently ordered. The latter may be seen as a generalization of the quantum switch, allowing a physical interpretation for pure two-slot superchannels. An immediate corollary is that purifiable bipartite processes cannot violate device-independent causal inequalities.

► BibTeX data

@article{Yokojima2021consequencesof,
  doi = {10.22331/q-2021-04-26-441},
  url = {https://doi.org/10.22331/q-2021-04-26-441},
  title = {Consequences of preserving reversibility in quantum superchannels},
  author = {Yokojima, Wataru and Quintino, Marco T{\'{u}}lio and Soeda, Akihito and Murao, Mio},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {441},
  month = apr,
  year = {2021}
}

► References

[1] A. S. Holevo, Quantum systems, channels, information: a mathematical introduction, vol. 16. Walter de Gruyter, 2012.
https://doi.org/10.1515/9783110273403

[2] M. M. Wilde, Quantum Information Theory. Cambridge University Press, 2 ed., 2017. arXiv:1106.1445 [quant-ph].
https://doi.org/10.1017/9781316809976
arXiv:1106.1445

[3] M. Wolf, ``Quantum Channels and Operations Guided Tour,'' 2012. http://www-m5.ma.tum.de/foswiki/pub/M5/Allgemeines/MichaelWolf/QChannelLecture.pdf.
http://www-m5.ma.tum.de/foswiki/pub/M5/Allgemeines/MichaelWolf/QChannelLecture.pdf

[4] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Transforming quantum operations: Quantum supermaps,'' EPL (Europhysics Letters) 83, 30004 (2008), arXiv:0804.0180 [quant-ph].
https://doi.org/10.1209/0295-5075/83/30004
arXiv:0804.0180

[5] K. Życzkowski, ``Quartic quantum theory: an extension of the standard quantum mechanics,'' J. Phys. A: Math. Gen. 41, 355302 (2008), arXiv:0804.1247 [quant-ph].
https://doi.org/10.1088/1751-8113/41/35/355302
arXiv:0804.1247

[6] O. Oreshkov, F. Costa, and Č. Brukner, ``Quantum correlations with no causal order,'' Nat. Commun. 3, 1092 (2012), arXiv:1105.4464 [quant-ph].
https://doi.org/10.1038/ncomms2076
arXiv:1105.4464

[7] M. Araújo, A. Feix, M. Navascués, and Č. Brukner, ``A purification postulate for quantum mechanics with indefinite causal order,'' Quantum 1, 10 (2017), arXiv:1611.08535 [quant-ph].
https://doi.org/10.22331/q-2017-04-26-10
arXiv:1611.08535

[8] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Quantum Circuit Architecture,'' Phys. Rev. Lett. 101, 060401 (2008), arXiv:0712.1325 [quant-ph].
https://doi.org/10.1103/PhysRevLett.101.060401
arXiv:0712.1325

[9] G. Chiribella, G. M. D'Ariano, P. Perinotti, and B. Valiron, ``Quantum computations without definite causal structure,'' Phys. Rev. A 88, 022318 (2013), arXiv:0912.0195 [quant-ph].
https://doi.org/10.1103/PhysRevA.88.022318
arXiv:0912.0195

[10] P. Perinotti, ``Causal Structures and the Classification of Higher Order Quantum Computations,'' in R. Renner and S. Stupar, eds., Time in Physics, pp. 103–127, Springer International Publishing, Cham, 2017, arXiv:1612.05099 [quant-ph].
https://doi.org/10.1007/978-3-319-68655-4_7
arXiv:1612.05099

[11] A. Bisio and P. Perinotti, ``Theoretical Framework for Higher-Order Quantum Theory,'' Proc. R. Soc. A 475, (2019), arXiv:1806.09554 [quant-ph].
https://doi.org/10.1098/rspa.2018.0706
arXiv:1806.09554

[12] A. Kissinger and S. Uijlen, ``A categorical semantics for causal structure,'' Logical Methods in Computer Science 15, (2019), arXiv:1701.04732 [quant-ph].
https://doi.org/10.23638/LMCS-15(3:15)2019
arXiv:1701.04732

[13] G. Chiribella, ``Perfect discrimination of no-signalling channels via quantum superposition of causal structures,'' Phys. Rev. A 86, 040301 (2012), arXiv:1109.5154 [quant-ph].
https://doi.org/10.1103/PhysRevA.86.040301
arXiv:1109.5154

[14] M. Araújo, F. Costa, and Č. Brukner, ``Computational Advantage from Quantum-Controlled Ordering of Gates,'' Phys. Rev. Lett. 113, 250402 (2014), arXiv:1401.8127 [quant-ph].
https://doi.org/10.1103/PhysRevLett.113.250402
arXiv:1401.8127

[15] P. A. Guérin, A. Feix, M. Araújo, and Č. Brukner, ``Exponential Communication Complexity Advantage from Quantum Superposition of the Direction of Communication,'' Phys. Rev. Lett. 117, 100502 (2016), arXiv:1605.07372 [quant-ph].
https://doi.org/10.1103/PhysRevLett.117.100502
arXiv:1605.07372

[16] D. Ebler, S. Salek, and G. Chiribella, ``Enhanced Communication with the Assistance of Indefinite Causal Order,'' Phys. Rev. Lett. 120, 120502 (2018), arXiv:1711.10165 [quant-ph].
https://doi.org/10.1103/PhysRevLett.120.120502
arXiv:1711.10165

[17] S. Salek, D. Ebler, and G. Chiribella, ``Quantum communication in a superposition of causal orders,'' arXiv:1809.06655 [quant-ph].
arXiv:1809.06655

[18] J. Bavaresco, M. Araújo, Č. Brukner, and M. T. Quintino, ``Semi-device-independent certification of indefinite causal order,'' Quantum 3, 176 (2019), arXiv:1903.10526 [quant-ph].
https://doi.org/10.22331/q-2019-08-19-176
arXiv:1903.10526

[19] M. Araújo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi, and Č. Brukner, ``Witnessing causal nonseparability,'' New Journal of Physics 17, 102001 (2015), arXiv:1506.03776 [quant-ph].
https://doi.org/10.1088/1367-2630/17/10/102001
arXiv:1506.03776

[20] M. T. Quintino, Q. Dong, A. Shimbo, A. Soeda, and M. Murao, ``Probabilistic exact universal quantum circuits for transforming unitary operations,'' Physical Review A 100, (2019), arXiv:1909.01366 [quant-ph].
https://doi.org/10.1103/physreva.100.062339
arXiv:1909.01366

[21] M. T. Quintino, Q. Dong, A. Shimbo, A. Soeda, and M. Murao, ``Reversing Unknown Quantum Transformations: Universal Quantum Circuit for Inverting General Unitary Operations,'' Physical Review Letters 123, (2019), arXiv:1810.06944 [quant-ph].
https://doi.org/10.1103/physrevlett.123.210502
arXiv:1810.06944

[22] L. M. Procopio, A. Moqanaki, M. Araújo, F. Costa, I. Alonso Calafell, E. G. Dowd, D. R. Hamel, L. A. Rozema, Č. Brukner, and P. Walther, ``Experimental superposition of orders of quantum gates,'' Nature Communications 6, 7913 (2015), arXiv:1412.4006 [quant-ph].
https://doi.org/10.1038/ncomms8913
arXiv:1412.4006

[23] G. Rubino, L. A. Rozema, A. Feix, M. Araújo, J. M. Zeuner, L. M. Procopio, v. Brukner, and P. Walther, ``Experimental verification of an indefinite causal order,'' Science Advances 3, e1602589 (2017), arXiv:1608.01683 [quant-ph].
https://doi.org/10.1126/sciadv.1602589
arXiv:1608.01683

[24] K. Goswami, C. Giarmatzi, M. Kewming, F. Costa, C. Branciard, J. Romero, and A. G. White, ``Indefinite Causal Order in a Quantum Switch,'' Phys. Rev. Lett. 121, 090503 (2018), arXiv:1803.04302 [quant-ph].
https://doi.org/10.1103/PhysRevLett.121.090503
arXiv:1803.04302

[25] O. Oreshkov, ``Time-delocalized quantum subsystems and operations: on the existence of processes with indefinite causal structure in quantum mechanics,'' Quantum 3, 206 (2019), arXiv:1801.07594 [quant-ph].
https://doi.org/10.22331/q-2019-12-02-206
arXiv:1801.07594

[26] P. Allard Guérin and Č. Brukner, ``Observer-dependent locality of quantum events,'' New Journal of Physics 20, 103031 (2018), arXiv:1805.12429 [quant-ph].
https://doi.org/10.1088/1367-2630/aae742
arXiv:1805.12429

[27] R. Lorenz and J. Barrett, ``Causal and compositional structure of unitary transformations,'' arXiv:2001.07774 [quant-ph].
arXiv:2001.07774

[28] J. Barrett, R. Lorenz, and O. Oreshkov, ``Cyclic quantum causal models,'' Nature Communications 12, (2021), arXiv:2002.12157 [quant-ph].
https://doi.org/10.1038/s41467-020-20456-x
arXiv:2002.12157

[29] J.-M. A. Allen, J. Barrett, D. C. Horsman, C. M. Lee, and R. W. Spekkens, ``Quantum Common Causes and Quantum Causal Models,'' Phys. Rev. X 7, 031021 (2017), arXiv:1609.09487 [quant-ph].
https://doi.org/10.1103/PhysRevX.7.031021
arXiv:1609.09487

[30] W. F. Stinespring, ``Positive Functions on C*-Algebras,'' Proceedings of the American Mathematical Society 6, 211–216 (1955).
https://doi.org/10.2307/2032342

[31] D. Kretschmann and R. F. Werner, ``Quantum channels with memory,'' Phys. Rev. A 72, 062323 (2005), quant-ph/0502106.
https://doi.org/10.1103/PhysRevA.72.062323
arXiv:quant-ph/0502106

[32] G. Gutoski and J. Watrous, ``Toward a general theory of quantum games,'' Proceedings of the thirty-ninth annual ACM symposium on Theory of computing - Proceedings of STOC 2007 , arXiv:quant-ph/0611234 [quant-ph].
https://doi.org/10.1145/1250790.1250873
arXiv:quant-ph/0611234

[33] A. Jamiołkowski, ``Linear transformations which preserve trace and positive semidefiniteness of operators,'' Reports on Mathematical Physics 3, 275–278 (1972).
https://doi.org/10.1016/0034-4877(72)90011-0

[34] M.-D. Choi, ``Completely positive linear maps on complex matrices,'' Linear algebra and its applications 10, 285–290 (1975).
https://doi.org/10.1016/0024-3795(75)90075-0

[35] G. Gour, ``Comparison of Quantum Channels by Superchannels,'' IEEE Transactions on Information Theory (2019), arXiv:1808.02607 [quant-ph].
https://doi.org/10.1109/tit.2019.2907989
arXiv:1808.02607

[36] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Theoretical framework for quantum networks,'' Phys. Rev. A 80, 022339 (2009), arXiv:0904.4483 [quant-ph].
https://doi.org/10.1103/PhysRevA.80.022339
arXiv:0904.4483

[37] A. Bisio, G. M. D'Ariano, P. Perinotti, and G. Chiribella, ``Minimal computational-space implementation of multiround quantum protocols,'' Phys. Rev. A 83, 022325 (2011), arXiv:1006.1780 [quant-ph].
https://doi.org/10.1103/PhysRevA.83.022325
arXiv:1006.1780

[38] J. Wechs, A. A. Abbott, and C. Branciard, ``On the definition and characterisation of multipartite causal (non)separability,'' New Journal of Physics 21, 013027 (2019), arXiv:1807.10557 [quant-ph].
https://doi.org/10.1088/1367-2630/aaf352
arXiv:1807.10557

[39] C. Branciard, M. Araújo, A. Feix, F. Costa, and Č. Brukner, ``The simplest causal inequalities and their violation,'' New Journal of Physics 18, 013008 (2015), arXiv:1508.01704 [quant-ph].
https://doi.org/10.1088/1367-2630/18/1/013008
arXiv:1508.01704

[40] O. Oreshkov and C. Giarmatzi, ``Causal and causally separable processes,'' New Journal of Physics 18, 093020 (2016), arXiv:1506.05449 [quant-ph].
https://doi.org/10.1088/1367-2630/18/9/093020
arXiv:1506.05449

[41] A. Feix, M. Araújo, and Č. Brukner, ``Causally nonseparable processes admitting a causal model,'' New Journal of Physics 18, 083040 (2016), arXiv:1604.03391 [quant-ph].
https://doi.org/10.1088/1367-2630/18/8/083040
arXiv:1604.03391

[42] M. M. Taddei, R. V. Nery, and L. Aolita, ``Quantum superpositions of causal orders as an operational resource,'' Physical Review Research 1, , arXiv:1903.06180 [quant-ph].
https://doi.org/10.1103/PhysRevResearch.1.033174
arXiv:1903.06180

[43] Ä. Baumeler and S. Wolf, ``The space of logically consistent classical processes without causal order,'' New Journal of Physics 18, 013036 (2016), arXiv:1507.01714 [quant-ph].
https://doi.org/10.1088/1367-2630/18/1/013036
arXiv:1507.01714

[44] M. Araújo, P. A. Guérin, and Ä. Baumeler, ``Quantum computation with indefinite causal structures,'' Physical Review A 96, 052315 (2017), arXiv:1706.09854 [quant-ph].
https://doi.org/10.1103/physreva.96.052315
arXiv:1706.09854

[45] P. Hayden, R. Jozsa, D. Petz, and A. Winter, ``Structure of states which satisfy strong subadditivity of quantum entropy with equality,'' Communications in mathematical physics 246, 359–374 (2004), arXiv:quant-ph/0304007.
https://doi.org/10.1007/s00220-004-1049-z
arXiv:quant-ph/0304007

[46] M. Koashi and N. Imoto, ``What is Possible Without Disturbing Partially Known Quantum States?,'' arXiv:quant-ph/0101144.
arXiv:quant-ph/0101144

[47] C. R. Fletcher, ``Elementary rings and modules, by Iain T. Adamson. Pp 136.\pounds 1\textperiodcentered 50. 1972 (Oliver and Boyd),'' The Mathematical Gazette 57, 145–145 (1973).
https://doi.org/10.1017/S0025557200132267

[48] P. R. Halmos, ``Finite-dimensional vector spaces,'' p. 30, Springer, 1958.
https://doi.org/10.1007/978-1-4612-6387-6

[49] T. S. Blyth and E. F. Robertson, ``Further linear algebra,'' p. 110, Springer, 2002.
https://doi.org/10.1007/978-1-4471-0661-6

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[1] Jonathan Barrett, Robin Lorenz, and Ognyan Oreshkov, "Cyclic quantum causal models", Nature Communications 12 1, 885 (2021).

[2] Julian Wechs, Hippolyte Dourdent, Alastair A. Abbott, and Cyril Branciard, "Quantum Circuits with Classical Versus Quantum Control of Causal Order", PRX Quantum 2 3, 030335 (2021).

[3] Jessica Bavaresco, Mio Murao, and Marco Túlio Quintino, "Unitary channel discrimination beyond group structures: Advantages of sequential and indefinite-causal-order strategies", Journal of Mathematical Physics 63 4, 042203 (2022).

[4] Julian Wechs, Cyril Branciard, and Ognyan Oreshkov, "Existence of processes violating causal inequalities on time-delocalised subsystems", Nature Communications 14 1, 1471 (2023).

[5] Fabio Costa, "A no-go theorem for superpositions of causal orders", Quantum 6, 663 (2022).

[6] Marco Túlio Quintino and Daniel Ebler, "Deterministic transformations between unitary operations: Exponential advantage with adaptive quantum circuits and the power of indefinite causality", Quantum 6, 679 (2022).

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[10] Tom Purves and Anthony J. Short, "Quantum Theory Cannot Violate a Causal Inequality", Physical Review Letters 127 11, 110402 (2021).

[11] Zbigniew Puchała, Kamil Korzekwa, Roberto Salazar, Paweł Horodecki, and Karol Życzkowski, "Dephasing superchannels", Physical Review A 104 5, 052611 (2021).

[12] Marco Túlio Quintino and Daniel Ebler, "Deterministic transformations between unitary operations: Exponential advantage with adaptive quantum circuits and the power of indefinite causality", arXiv:2109.08202, (2021).

[13] Simon Milz and Marco Túlio Quintino, "Transformations between arbitrary (quantum) objects and the emergence of indefinite causality", arXiv:2305.01247, (2023).

[14] Augustin Vanrietvelde, Nick Ormrod, Hlér Kristjánsson, and Jonathan Barrett, "Consistent circuits for indefinite causal order", arXiv:2206.10042, (2022).

[15] Kaumudibikash Goswami and Fabio Costa, "Classical communication through quantum causal structures", Physical Review A 103 4, 042606 (2021).

[16] Qingxiuxiong Dong, Marco Túlio Quintino, Akihito Soeda, and Mio Murao, "The quantum switch is uniquely defined by its action on unitary operations", arXiv:2106.00034, (2021).

[17] Simon Milz, Dominic Jurkschat, Felix A. Pollock, and Kavan Modi, "Delayed-choice causal order and nonclassical correlations", Physical Review Research 3 2, 023028 (2021).

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