Characterising transformations between quantum objects, ‘completeness’ of quantum properties, and transformations without a fixed causal order
1School of Physics, Trinity College Dublin, Dublin 2, Ireland
2Trinity Quantum Alliance, Unit 16, Trinity Technology and Enterprise Centre, Pearse Street, Dublin 2, D02YN67, Ireland
3Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
4Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria
5Sorbonne Université, CNRS, LIP6, F-75005 Paris, France
Published: | 2024-07-17, volume 8, page 1415 |
Eprint: | arXiv:2305.01247v2 |
Doi: | https://doi.org/10.22331/q-2024-07-17-1415 |
Citation: | Quantum 8, 1415 (2024). |
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Abstract
Many fundamental and key objects in quantum mechanics are linear mappings between particular affine/linear spaces. This structure includes basic quantum elements such as states, measurements, channels, instruments, non-signalling channels and channels with memory, and also higher-order operations such as superchannels, quantum combs, n-time processes, testers, and process matrices which may not respect a definite causal order. Deducing and characterising their structural properties in terms of linear and semidefinite constraints is not only of foundational relevance, but plays an important role in enabling the numerical optimisation over sets of quantum objects and allowing simpler connections between different concepts and objects. Here, we provide a general framework to deduce these properties in a direct and easy to use way. While primarily guided by practical quantum mechanical considerations, we also extend our analysis to mappings between $general$ linear/affine spaces and derive their properties, opening the possibility for analysing sets which are not explicitly forbidden by quantum theory, but are still not much explored. Together, these results yield versatile and readily applicable tools for all tasks that require the characterisation of linear transformations, in quantum mechanics and beyond. As an application of our methods, we discuss how the existence of indefinite causality naturally emerges in higher-order quantum transformations and provide a simple strategy for the characterisation of mappings that have to preserve properties in a 'complete' sense, i.e., when acting non-trivially only on parts of an input space.
![](https://quantum-journal.org/wp-content/uploads/2024/07/QuantumFig.png)
Featured image: A pictorial illustration of an exemplary chain of higher-order operations. Channels describe transformations between states, superchannels describe transformation between channels, and supersuperchannels describe transformations between superchannels. The methods we provide allow for the characterization of all mappings in this chain, but more generally all mappings between arbitrary quantum objects.
Popular summary
The crucial role played by transformations in quantum mechanics is already visible when considering its basic building blocks. To name but two pertinent examples, a quantum evolution — described by quantum channels — transforms quantum states at an earlier time to quantum states at a later time, while quantum measurements transform quantum states to outcome probabilities. Taking this role played by mappings between different sets of quantum objects as a starting point, one does not have to stop at quantum states, channels and measurements, but can, for example consider seemingly more exotic transformations, like those that map quantum channels onto quantum channels, or, say, pairs of quantum channels to quantum states.
Indeed, many of the resulting, so-called higher order transformations, like, e.g., quantum combs or quantum testers, turn out to be compatible with standard quantum mechanics and have found ample application in quantum information theory. Others, like, e.g., process matrices or the quantum time flip cannot necessarily be implemented within quantum mechanics when assuming a fixed causal order, but allow one to investigate the potential consequences of causal indefiniteness in quantum mechanics. Studying the hierarchy of transformations between sets of quantum objects thus promises a whole host of interesting higher-order transformations.
The present work rests agnostic of their respective physicality and offers a systematic way to derive the pertinent properties of transformations between arbitrary sets of quantum objects, thus providing the basic toolbox for this study.
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► References
[1] G. Chiribella, G. M. D'Ariano, and P. Perinotti, Transforming quantum operations: Quantum supermaps, EPL 83, 30004 (2008a), arXiv:0804.0180.
https://doi.org/10.1209/0295-5075/83/30004
arXiv:0804.0180
[2] G. Chiribella, G. M. D'Ariano, and P. Perinotti, Quantum Circuit Architecture, Phys. Rev. Lett. 101, 060401 (2008b), arXiv:0712.1325.
https://doi.org/10.1103/PhysRevLett.101.060401
arXiv:0712.1325
[3] 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 (2007) pp. 565–574, arXiv:quant-ph/0611234.
https://doi.org/10.1145/1250790.1250873
arXiv:quant-ph/0611234
[4] 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 (2018), arXiv:1512.00589.
https://doi.org/10.1103/PhysRevA.97.012127
arXiv:1512.00589
[5] G. Chiribella, G. M. D'Ariano, and P. Perinotti, Theoretical framework for quantum networks, Phys. Rev. A 80, 022339 (2009), arXiv:0904.4483.
https://doi.org/10.1103/PhysRevA.80.022339
arXiv:0904.4483
[6] M. Ziman, Process positive-operator-valued measure: A mathematical framework for the description of process tomography experiments, Phys. Rev. A 77, 062112 (2008), arXiv:0802.3862.
https://doi.org/10.1103/PhysRevA.77.062112
arXiv:0802.3862
[7] J. Bavaresco, M. Murao, and M. T. Quintino, Strict Hierarchy between Parallel, Sequential, and Indefinite-Causal-Order Strategies for Channel Discrimination, Phys. Rev. Lett. 127, 200504 (2021), arXiv:2011.08300.
https://doi.org/10.1103/PhysRevLett.127.200504
arXiv:2011.08300
[8] O. Oreshkov, F. Costa, and Č. Brukner, Quantum correlations with no causal order, Nat. Commun. 3, 1092 (2012), arXiv:1105.4464.
https://doi.org/10.1038/ncomms2076
arXiv:1105.4464
[9] D. Kretschmann and R. F. Werner, Quantum channels with memory, Phys. Rev. A 72, 062323 (2005), arXiv:quant-ph/0502106.
https://doi.org/10.1103/PhysRevA.72.062323
arXiv:quant-ph/0502106
[10] E. Castro-Ruiz, F. Giacomini, and Č. Brukner, Dynamics of Quantum Causal Structures, Phys. Rev. X 8, 011047 (2018), arXiv:1710.03139.
https://doi.org/10.1103/PhysRevX.8.011047
arXiv:1710.03139
[11] P. Perinotti, Causal structures and the classification of higher order quantum computations, Tutorials, Schools, and Workshops in the Mathematical Sciences , 103–127 (2017), arXiv:1612.05099.
https://doi.org/10.1007/978-3-319-68655-4_7
arXiv:1612.05099
[12] A. Bisio and P. Perinotti, Theoretical framework for higher-order quantum theory, Proc. R. Soc. A 475, 20180706 (2019), arXiv:1806.09554.
https://doi.org/10.1098/rspa.2018.0706
arXiv:1806.09554
[13] W. Simmons and A. Kissinger, Higher-order causal theories are models of BV-logic, arXiv:2205.11219 (2022).
arXiv:2205.11219
[14] T. Hoffreumon and O. Oreshkov, Projective characterization of higher-order quantum transformations, arXiv e-prints (2022).
arXiv:2206.06206v2
[15] M. Araújo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi, and Č. Brukner, Witnessing causal nonseparability, New J. Phys. 17, 102001 (2015), arXiv:1506.03776.
https://doi.org/10.1088/1367-2630/17/10/102001
arXiv:1506.03776
[16] S. Milz, J. Bavaresco, and G. Chiribella, Resource theory of causal connection, Quantum 6, 788 (2022), arXiv:2110.03233.
https://doi.org/10.22331/q-2022-08-25-788
arXiv:2110.03233
[17] L. Apadula, A. Bisio, and P. Perinotti, No-signalling constrains quantum computation with indefinite causal structure, Quantum 8, 1241 (2024), arXiv:2202.10214.
https://doi.org/10.22331/q-2024-02-05-1241
arXiv:2202.10214
[18] 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.
https://doi.org/10.22331/q-2017-04-26-10
arXiv:1611.08535
[19] J. Burniston, M. Grabowecky, C. M. Scandolo, G. Chiribella, and G. Gour, Necessary and sufficient conditions on measurements of quantum channels, Proc. R. Soc. A 476, 20190832 (2020), arXiv:1904.09161.
https://doi.org/10.1098/rspa.2019.0832
arXiv:1904.09161
[20] M. Wilson, G. Chiribella, and A. Kissinger, Quantum Supermaps are Characterized by Locality, arXiv e-prints (2023), arXiv:2205.09844 [quant-ph].
arXiv:2205.09844
[21] M. Wilson and N. Ormrod, On the Origin of Linearity and Unitarity in Quantum Theory, arXiv e-prints (2023), arXiv:2305.20063 [quant-ph].
arXiv:2305.20063
[22] J. de Pillis, Linear transformations which preserve hermitian and positive semidefinite operators, Pac. J. Math. 23, 129–137 (1967).
https://doi.org/10.2140/pjm.1967.23.129
[23] A. Jamiołkowski, Linear transformations which preserve trace and positive semidefiniteness of operators, Rep. Math. Phys. 3, 275–278 (1972).
https://doi.org/10.1016/0034-4877(72)90011-0
[24] M.-D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl. 10, 285 – 290 (1975).
https://doi.org/10.1016/0024-3795(75)90075-0
[25] S. Milz and K. Modi, Quantum Stochastic Processes and Quantum non-Markovian Phenomena, PRX Quantum 2, 030201 (2021), arXiv:2012.01894.
https://doi.org/10.1103/PRXQuantum.2.030201
arXiv:2012.01894
[26] D. Beckman, D. Gottesman, M. A. Nielsen, and J. Preskill, Causal and localizable quantum operations, Phys. Rev. A 64, 052309 (2001), arXiv:quant-ph/0102043.
https://doi.org/10.1103/PhysRevA.64.052309
arXiv:quant-ph/0102043
[27] M. Piani, M. Horodecki, P. Horodecki, and R. Horodecki, Properties of quantum nonsignaling boxes, Phys. Rev. A 74, 012305 (2006), arXiv:quant-ph/0505110.
https://doi.org/10.1103/PhysRevA.74.012305
arXiv:quant-ph/0505110
[28] K. Modi, Operational approach to open dynamics and quantifying initial correlations, Sci. Rep. 2, 581 (2012), arXiv:1011.6138.
https://doi.org/10.1038/srep00581
arXiv:1011.6138
http://www.nature.com/articles/srep00581
[29] G. Chiribella, G. M. D’Ariano, and P. Perinotti, Theoretical framework for quantum networks, Phys. Rev. A 80, 022339 (2009), arXiv:0904.4483.
https://doi.org/10.1103/PhysRevA.80.022339
arXiv:0904.4483
[30] M. T. Quintino and D. Ebler, Deterministic transformations between unitary operations: Exponential advantage with adaptive quantum circuits and the power of indefinite causality, Quantum 6, 679 (2022), arXiv:2109.08202.
https://doi.org/10.22331/q-2022-03-31-679
arXiv:2109.08202
[31] O. Oreshkov and C. Giarmatzi, Causal and causally separable processes, New J. Phys. 18, 093020 (2016), arXiv:1506.05449.
https://doi.org/10.1088/1367-2630/18/9/093020
arXiv:1506.05449
[32] J. Wechs, A. A. Abbott, and C. Branciard, On the definition and characterisation of multipartite causal (non)separability, New J. Phys. 21, 013027 (2019), arXiv:1807.10557.
https://doi.org/10.1088/1367-2630/aaf352
arXiv:1807.10557
[33] J. Wechs, H. Dourdent, A. A. Abbott, and C. Branciard, Quantum Circuits with Classical Versus Quantum Control of Causal Order, PRX Quantum 2, 030335 (2021), arXiv:2101.08796.
https://doi.org/10.1103/PRXQuantum.2.030335
arXiv:2101.08796
[34] G. Gutoski, Properties of local quantum operations with shared entanglement, Quantum Inf. Comput. 9, 739–764 (2009), arXiv:0805.2209.
arXiv:0805.2209
https://dl.acm.org/doi/10.5555/2011804.2011806
[35] 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.
https://doi.org/10.1103/PhysRevA.88.022318
arXiv:0912.0195
[36] C. Branciard, M. Araújo, A. Feix, F. Costa, and Č. Brukner, The simplest causal inequalities and their violation, New J. Phys. 18, 013008 (2015), arXiv:1508.01704.
https://doi.org/10.1088/1367-2630/18/1/013008
arXiv:1508.01704
[37] G. Chiribella and Z. Liu, Quantum operations with indefinite time direction, Commun. Phys. 5, 190 (2022), arXiv:2012.03859.
https://doi.org/10.1038/s42005-022-00967-3
arXiv:2012.03859
[38] T. Strömberg, P. Schiansky, M. T. Quintino, M. Antesberger, L. A. Rozema, I. Agresti, Č. Brukner, and P. Walther, Experimental superposition of a quantum evolution with its time reverse, Phys. Rev. Research 6, 023071 (2024), arXiv:2211.01283.
https://doi.org/10.1103/PhysRevResearch.6.023071
arXiv:2211.01283
[39] Y. Guo, Z. Liu, H. Tang, X.-M. Hu, B.-H. Liu, Y.-F. Huang, C.-F. Li, G.-C. Guo, and G. Chiribella, Experimental Demonstration of Input-Output Indefiniteness in a Single Quantum Device, Phys. Rev. Lett. 132, 160201 (2024), arXiv:2210.17046.
https://doi.org/10.1103/PhysRevLett.132.160201
arXiv:2210.17046
[40] P. Faist, J. Oppenheim, and R. Renner, Gibbs-preserving maps outperform thermal operations in the quantum regime, New J. Phys. 17, 043003 (2015), arXiv:1406.3618.
https://doi.org/10.1088/1367-2630/17/4/043003
arXiv:1406.3618
[41] M. Lostaglio, An introductory review of the resource theory approach to thermodynamics, Rep. Prog. Phys. 82, 114001 (2019), arXiv:1807.11549.
https://doi.org/10.1088/1361-6633/ab46e5
arXiv:1807.11549
[42] E. B. Davis, Quantum Theory of Open Systems (Academic Press Inc, London; New York, 1976).
[43] G. Lindblad, Non-Markovian quantum stochastic processes and their entropy, Comm. Math. Phys. 65, 281–294 (1979).
https://doi.org/10.1007/BF01197883
[44] G. Chiribella and D. Ebler, Optimal quantum networks and one-shot entropies, New J. Phys. 18, 093053 (2016), arXiv:1606.02394.
https://doi.org/10.1088/1367-2630/18/9/093053
arXiv:1606.02394
[45] 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.
https://doi.org/10.22331/q-2019-08-19-176
arXiv:1903.10526
[46] P. Lewandowska, L. Pawela, and Z. Puchała, Strategies for single-shot discrimination of process matrices, Sci. Rep. 13, 3046 (2023), arXiv:2210.14575.
https://doi.org/10.1038/s41598-023-30191-0
arXiv:2210.14575
[47] W. Yokojima, M. T. Quintino, A. Soeda, and M. Murao, Consequences of preserving reversibility in quantum superchannels, Quantum 5, 441 (2021), arXiv:2003.05682.
https://doi.org/10.22331/q-2021-04-26-441
arXiv:2003.05682
[48] Ä. Baumeler and S. Wolf, The space of logically consistent classical processes without causal order, New J. of Phys. 18, 013036 (2016), arXiv:1507.01714.
https://doi.org/10.1088/1367-2630/18/1/013036
arXiv:1507.01714
[49] A. Vanrietvelde, N. Ormrod, H. Kristjánsson, and J. Barrett, Consistent circuits for indefinite causal order, arXiv:2206.10042 (2022).
arXiv:2206.10042
[50] C. Branciard, Witnesses of causal nonseparability: an introduction and a few case studies, Sci. Rep. 6, 26018 (2016), arXiv:1603.00043.
https://doi.org/10.1038/srep26018
arXiv:1603.00043
[51] M. Nery, M. T. Quintino, P. A. Guérin, T. O. Maciel, and R. O. Vianna, Simple and maximally robust processes with no classical common-cause or direct-cause explanation, Quantum 5, 538 (2021), arXiv:2101.11630.
https://doi.org/10.22331/q-2021-09-09-538
arXiv:2101.11630
[52] 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 (2019), arXiv:1909.01366.
https://doi.org/10.1103/PhysRevA.100.062339
arXiv:1909.01366
[53] S. Yoshida, A. Soeda, and M. Murao, Universal construction of decoders from encoding black boxes, Quantum 7, 957 (2023a), arXiv:2110.00258.
https://doi.org/10.22331/q-2023-03-20-957
arXiv:2110.00258
[54] S. Yoshida, A. Soeda, and M. Murao, Reversing Unknown Qubit-Unitary Operation, Deterministically and Exactly, Phys. Rev. Lett. 131, 120602 (2023b), arXiv:2209.02907.
https://doi.org/10.1103/PhysRevLett.131.120602
arXiv:2209.02907
[55] J. Bavaresco, M. Murao, and M. T. Quintino, Unitary channel discrimination beyond group structures: Advantages of sequential and indefinite-causal-order strategies, J. Math. Phys. 63, 042203 (2022), arXiv:2105.13369.
https://doi.org/10.1063/5.0075919
arXiv:2105.13369
[56] M. T. Quintino, https://github.com/mtcq/higherorderprojectors (2023).
https://github.com/mtcq/HigherOrderProjectors
[57] G. Gour, Comparison of Quantum Channels by Superchannels, IEEE Trans. Inf. Theory 65, 5880–5904 (2019), arXiv:1808.02607.
https://doi.org/10.1109/TIT.2019.2907989
arXiv:1808.02607
[58] G. Gour, D. Kim, T. Nateeboon, G. Shemesh, and G. Yoeli, Inevitable Negativity: Additivity Commands Negative Quantum Channel Entropy, arXiv e-prints (2024), arXiv:2406.13823 [quant-ph].
arXiv:2406.13823
[59] A. Kissinger and S. Uijlen, A categorical semantics for causal structure, Log. Methods Comput. Sci. 15, 3 (2019), arXiv:1701.04732.
https://doi.org/10.23638/LMCS-15(3:15)2019
arXiv:1701.04732
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[2] Jessica Bavaresco, Patryk Lipka-Bartosik, Pavel Sekatski, and Mohammad Mehboudi, "Designing optimal protocols in Bayesian quantum parameter estimation with higher-order operations", Physical Review Research 6 2, 023305 (2024).
[3] Guilherme Zambon and Diogo O. Soares-Pinto, "Relations between Markovian and non-Markovian correlations in multitime quantum processes", Physical Review A 109 6, 062401 (2024).
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