The quantum switch is uniquely defined by its action on unitary operations

Qingxiuxiong Dong; Marco Túlio Quintino; Akihito Soeda; Mio Murao

doi:10.22331/q-2023-11-07-1169

The quantum switch is uniquely defined by its action on unitary operations

Qingxiuxiong Dong¹, Marco Túlio Quintino^2,3,4,1, Akihito Soeda^5,6,1, and Mio Murao^1,7

¹Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
²Sorbonne Université, CNRS, LIP6, F-75005 Paris, France
³Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria
⁴Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
⁵Principles of Informatics Research Division, National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku,Tokyo 101-8430, Japan
⁶Department of Informatics, School of Multidisciplinary Sciences, SOKENDAI (The Graduate University for Advanced Studies), 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
⁷Trans-scale Quantum Science Institute, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

Published:	2023-11-07, volume 7, page 1169
Eprint:	arXiv:2106.00034v5
Doi:	https://doi.org/10.22331/q-2023-11-07-1169
Citation:	Quantum 7, 1169 (2023).

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Abstract

The quantum switch is a quantum process that creates a coherent control between different unitary operations, which is often described as a quantum process which transforms a pair of unitary operations $(U_1 , U_2)$ into a controlled unitary operation that coherently applies them in different orders as $\vert {0} \rangle\langle {0} \vert \otimes U_1 U_2 + \vert {1} \rangle\langle {1} \vert \otimes U_2 U_1$. This description, however, does not directly define its action on non-unitary operations. The action of the quantum switch on non-unitary operations is then chosen to be a ``natural'' extension of its action on unitary operations. In general, the action of a process on non-unitary operations is not uniquely determined by its action on unitary operations. It may be that there could be a set of inequivalent extensions of the quantum switch for non-unitary operations. We prove, however, that the natural extension is the only possibility for the quantum switch for the 2-slot case. In other words, contrary to the general case, the action of the quantum switch on non-unitary operations (as a linear and completely CP preserving supermap) is completely determined by its action on unitary operations. We also discuss the general problem of when the complete description of a quantum process is uniquely determined by its action on unitary operations and identify a set of single-slot processes which are completely defined by their action on unitary operations.

Featured image: If a two-slot superchannel on every pair of unitary channels as the quantum switch, then it acts on any pair of quantum operations as the quantum switch. In other words, the quantum switch is uniquely defined by its action on unitary channels.

► BibTeX data

@article{Dong2023quantumswitchis,
  doi = {10.22331/q-2023-11-07-1169},
  url = {https://doi.org/10.22331/q-2023-11-07-1169},
  title = {The quantum switch is uniquely defined by its action on unitary operations},
  author = {Dong, Qingxiuxiong 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 = {7},
  pages = {1169},
  month = nov,
  year = {2023}
}

► References

[1] M. Ozawa, Conditional expectation and repeated measurements of continuous quantum observables, Springer, Berlin, Heidelberg https://doi.org/10.1007/BFb0072947 (1983).
https://doi.org/10.1007/BFb0072947

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

[3] 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

[4] 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

[5] 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

[6] 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

[7] 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

[8] 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

[9] S. Salek, D. Ebler, and G. Chiribella, Quantum communication in a superposition of causal orders, arXiv e-prints (2018), arXiv:1809.06655 [quant-ph].
arXiv:1809.06655

[10] G. Chiribella, M. Banik, S. Sankar Bhattacharya, T. Guha, M. Alimuddin, A. Roy, S. Saha, S. Agrawal, and G. Kar, Indefinite causal order enables perfect quantum communication with zero capacity channel, arXiv e-prints (2018), arXiv:1810.10457 [quant-ph].
https://doi.org/10.1088/1367-2630/abe7a0
arXiv:1810.10457

[11] K. Goswami, Y. Cao, G. A. Paz-Silva, J. Romero, and A. G. White, Increasing communication capacity via superposition of order, Phys. Rev. Research 2, 033292 (2020), arXiv:1807.07383 [quant-ph].
https://doi.org/10.1103/PhysRevResearch.2.033292
arXiv:1807.07383

[12] Y. Guo, X.-M. Hu, Z.-B. Hou, H. Cao, J.-M. Cui, B.-H. Liu, Y.-F. Huang, C.-F. Li, G.-C. Guo, and G. Chiribella, Experimental Transmission of Quantum Information Using a Superposition of Causal Orders, Phys. Rev. Lett. 124, 030502 (2020), arXiv:1811.07526 [quant-ph].
https://doi.org/10.1103/PhysRevLett.124.030502
arXiv:1811.07526

[13] K. Kraus, States, effects and operations: fundamental notions of quantum theory, Springer https://doi.org/10.1007/3-540-12732-1 (1983).
https://doi.org/10.1007/3-540-12732-1

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

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

[16] 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

[17] A. Chefles, Deterministic quantum state transformations, Physics Letters A 270, 14–19 (2000), arXiv:quant-ph/9911086 [quant-ph].
https://doi.org/10.1016/S0375-9601(00)00291-7
arXiv:quant-ph/9911086

[18] A. Chefles, R. Jozsa, and A. Winter, On the existence of physical transformations between sets of quantum states, International Journal of Quantum Information 02, 11–21 (2004), quant-ph/0307227.
https://doi.org/10.1142/s0219749904000031
arXiv:quant-ph/0307227

[19] T. Heinosaari, M. A. Jivulescu, D. Reeb, and M. M. Wolf, Extending quantum operations, Journal of Mathematical Physics 53, 102208–102208 (2012), arXiv:1205.0641 [math-ph].
https://doi.org/10.1063/1.4755845
arXiv:1205.0641

[20] Q. Dong, M. T. Quintino, A. Soeda, and M. Murao, Implementing positive maps with multiple copies of an input state, Phys. Rev. A 99, 052352 (2019), arXiv:1808.05788 [quant-ph].
https://doi.org/10.1103/PhysRevA.99.052352
arXiv:1808.05788

[21] E. Haapasalo, T. Heinosaari, and J. P. Pellonpaa, When do pieces determine the whole? Extreme marginals of a completely positive map, Reviews in Mathematical Physics 26, 1450002 (2014), arXiv:1209.5933 [quant-ph].
https://doi.org/10.1142/S0129055X14500020
arXiv:1209.5933

[22] L. Guerini and M. Terra Cunha, Uniqueness of the joint measurement and the structure of the set of compatible quantum measurements, Journal of Mathematical Physics 59, 042106 (2018), arXiv:1711.04804 [quant-ph].
https://doi.org/10.1063/1.5017699
arXiv:1711.04804

[23] J. Barrett, Information processing in generalized probabilistic theories, Phys. Rev. A 75, 032304 (2007), arXiv:quant-ph/0508211 [quant-ph].
https://doi.org/10.1103/PhysRevA.75.032304
arXiv:quant-ph/0508211

[24] 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

[25] 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

[26] 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

[27] W. Yokojima, M. T. Quintino, A. Soeda, and M. Murao, Consequences of preserving reversibility in quantum superchannels, Quantum 5, 441 (2021), 2003.05682.
https://doi.org/10.22331/q-2021-04-26-441
arXiv:2003.05682

[28] M. T. Quintino, Q. Dong, A. Shimbo, A. Soeda, and M. Murao, Probabilistic exact universal quantum circuits for transforming unitary operations, Phys. Rev. A 100, 062339 (2019a), arXiv:1909.01366 [quant-ph].
https://doi.org/10.1103/PhysRevA.100.062339
arXiv:1909.01366

[29] Q. Dong, M. T. Quintino, A. Soeda, and M. Murao, Success-or-Draw: A Strategy Allowing Repeat-Until-Success in Quantum Computation, Phys. Rev. Lett. 126, 150504 (2021), arXiv:2011.01055 [quant-ph].
https://doi.org/10.1103/PhysRevLett.126.150504
arXiv:2011.01055

[30] J. Miyazaki, A. Soeda, and M. Murao, Complex conjugation supermap of unitary quantum maps and its universal implementation protocol, Phys. Rev. Research 1, arXiv:1706.03481 (2019), arXiv:1706.03481 [quant-ph].
https://doi.org/10.1103/PhysRevResearch.1.013007
arXiv:1706.03481

[31] M. T. Quintino, Q. Dong, A. Shimbo, A. Soeda, and M. Murao, Reversing unknown quantum transformations: A universal quantum circuit for inverting general unitary operations, Phys. Rev. Lett. 123, 210502 (2019b), arXiv:1810.06944 [quant-ph].
https://doi.org/10.1103/PhysRevLett.123.210502
arXiv:1810.06944

[32] G. Chiribella and Z. Liu, Quantum operations with indefinite time direction, arXiv e-prints (2020), arXiv:2012.03859 [quant-ph].
https://doi.org/10.1038/s42005-022-00967-3
arXiv:2012.03859

[33] J. Barrett, R. Lorenz, and O. Oreshkov, Cyclic quantum causal models, Nature Communications 12, 885 (2021), arXiv:2002.12157 [quant-ph].
https://doi.org/10.1038/s41467-020-20456-x
arXiv:2002.12157

[34] J. Wechs, H. Dourdent, A. A. Abbott, and C. Branciard, Quantum circuits with classical versus quantum control of causal order, arXiv e-prints (2021), arXiv:2101.08796 [quant-ph].
https://doi.org/10.1103/PRXQuantum.2.030335
arXiv:2101.08796

[35] R. A. Bertlmann and P. Krammer, Bloch vectors for qudits, Journal of Physics A Mathematical General 41, 235303 (2008), arXiv:0806.1174 [quant-ph].
https://doi.org/10.1088/1751-8113/41/23/235303
arXiv:0806.1174

[36] C. B. Mendl and M. M. Wolf, Unital quantum channels – convex structure and revivals of birkhoff’s theorem, Communications in Mathematical Physics 289, 1057–1086 (2009), arXiv:0806.2820 [quant-ph].
https://doi.org/10.1007/s00220-009-0824-2
arXiv:0806.2820

Cited by

[1] Giulio Chiribella and Zixuan Liu, "Quantum operations with indefinite time direction", Communications Physics 5 1, 190 (2022).

[2] 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).

[3] Satoshi Yoshida, Akihito Soeda, and Mio Murao, "Universal construction of decoders from encoding black boxes", arXiv:2110.00258, (2021).

[4] Matt Wilson and Giulio Chiribella, "Causality in Higher Order Process Theories", arXiv:2107.14581, (2021).

[5] Satoshi Yoshida, Akihito Soeda, and Mio Murao, "Universal construction of decoders from encoding black boxes", Quantum 7, 957 (2023).

The above citations are from SAO/NASA ADS (last updated successfully 2023-12-07 04:39:29). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2023-12-07 04:39:27).

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